How do you ace honor physics and trigonometry, yet struggle to understand algebra? High school physics and trig are basically just coming up with the correct algebraic equation, and then solving it. (Or just turn the crank, my physics teacher always said.) I can understand not knowing calculus for simple physics, since you can just approximate it with the slope of the curve or area under the curve, but that still takes algebra. Trigonometry is just using the various theorems and identities, but even using the Pythagorean theorem takes a little algebra to isolate the variable you want.
I admit math is very easy for me, but it blows my mind that people don't understand algebra at least.
Personally, I've always found algebra the most difficult - it's just symbol soup with no apparent concepts to latch onto. Perhaps it's just the way it's usually presented, but I don't "get it" at all. It's still by far the most daunting part of math for me, even after taking 5 math courses in university.
For me, algebra made a lot more sense after doing proofs in geometry class, and really pedantically showing what was going on. Before that, it was all "moving symbols around", which was slightly mystical, and more an exercise in doing what you're told than in figuring things out.
The key insight, which maybe people think is too obvious to say out loud, but which does need to be said out loud, is that if two quantities are the same, then they remain the same if you do the same thing to both of them. For example, if x = y, then also x - 3 = y - 3. So the entire game of algebra is to choose things to do identically to both sides of an equation. Anything about "moving" symbols is a shorthand for this process. For example, if you know "x = y + 3", then, yes, you do also know that "x - 3 = y", and someone may speak of "moving the three", but fundamentally what you're actually doing is not "moving" anything, but rather subtracting three from both the quantity on the left, and from the quantity on the right. You picked an operation to perform on both quantities, to get another true statement.
The exercise that finally made this make sense, was to painstakingly walk through simple proofs, in tiny tiny steps, starting from small numbers of self-evident postulates. This is in the tradition of Euclid's Elements, which has a totally different vibe from most math education. It was important to do this so slowly and pedantically that you would be embarrassed to do it -- you would worry people would think you are stupid -- if the correct norms had not been established. You need to establish the norm that leaping ahead and skipping steps is not a sign of intelligence but is instead sloppiness, ugliness, something missing, a flaw in the argument you're constructing.
It's interesting. I love math, and it's always been super easy for me. Yet ask me to play an instrument or paint a painting, and I'd have an easier time flying to Mars.
I am always surprised when I hear how americans students learn "Algebra" in highschool. It makes me think that German schools are way behind. I mean algebra is about mathematical structures like Fields, Rings, Groups or even "Algebras" (https://en.m.wikipedia.org/wiki/Algebra_over_a_field).
In American schools we what we call 'algebra' is just doing basic symbolic computation, like solving very simple equations.
The kind of algebra you're describing was called 'abstract algebra' at my university, and was an advanced pair of courses for math majors. The first semester covered group theory and the second semester covered ring theory.
Are German schools behind? In North America, learning algebra means we learn to solve for x/y : solving quadratic equations is pretty much the ceiling for "algebra" and even that isn't taught until grade 8, usually grade 9.
American math standards have been low compared to those in Europe and East Asia for a long time, about as long as there has been divergence in student performance in the subject. Standards should be high, and many students (perhaps 80%) should fail to meet them. But what does this matter if only 1/3 of Americans get undergraduate degrees anyway? I can tell you what isn’t working: expecting very little of American kids in the public school system. It’s harmful to the education of our youth and gives further advantage to the already privileged who can afford private schooling and tutoring.
Professors blame the pandemic, but the true root cause is the blithe attitude American society has towards mathematics that seems to be ubiquitous outside of the east coast. This affects each student a decade before they enter high school; according to Common Core (no citation because mobile) fractions are taught not until FOURTH GRADE. That is almost 3 years too late.
I wonder what it will take to change this. What went wrong, or what do other countries do that isn't done here? Why are we like this?
> fractions are taught not until FOURTH GRADE. That is almost 3 years too late.
Former math educator: This is not the problem at all. Finland has great Math Education and they wait till 5th grade. If you ever try teaching a large number of 1st graders (or 3rd graders) fractions it becomes clear most of them are not developmentally ready and they'd be better served firming up foundations that would let them more quickly learn fractions when they're ready.
More pertinent problems are that much of our k-6 teaching staff are themselves math illiterate, our culture of disliking mathematics is enforced by a teaching style that makes math incredibly stressful for kids and disengage from the subject, we don't allow any tracking of where students actually are which causes slow students to fall behind more and fast students to be bored out of their mind, and that we make no use of "developmental priming" to speed up later learning later (e.g. you can learn many of the concepts of calculus long before your brain is capable of the necessary algebraic operations, if you know them, you can learn calculus faster)
The PLATO system for so-called "programmed instruction" was available in the 1970s, IIRC. Meanwhile, math pedagogy, AFAIK, tends to more rigorous reference to "scope and sequence". These seem compatible even without AI, and if only to better identify gaps.
Turing and Gödel proving that whenever a mathematical system is rich enough to describe the arithmetic we learn at school, it cannot prove its own consistency is a bit of a barrier.
ML is sophisticated pattern matching and finding.
As we know that even defining simple rules of arithmetic is impossible, how will pattern finding do so.
Sure some company could pay slave wages to make LLMs better at it, but correctness is important in math and LLMs have no concept of truthfulness.
Feed forward Neural Networks, like LLMs using attention are effectively Directed acyclic graphs, so it has little it can do to 'evaluate from the outside'
Even if you can move to second-order logic logically-valid formulas in second-order logic is not recursively enumerable. AS RE problems are those which a Turing machine can answer "YES" in a finite amount of time, but a "NO" answer might never come, moving out of RE spaces like DAGs is not computationally feasible.
The problem with 'pattern finding' is that it can learn some even HoL problems if it is in the corpus, but will fail on even simpler problems that weren't.
"A persistent problem in corpus-based ML, in all its applications, is that the patterns that the AI finds do not actually reflect the fundamental characteristic of the problem, but rather superficial regularities in the training data, known as “artifacts”.
LLMs aren't doing the 'logic' of the math problems, they are finding patterns in their training data that are hopefully close enough to work for the presented problem.
This is why you can use AI to say learn about intervals on the real line, as those are of finite VC dimensionality, but algebra questions that are outside of it's corpus tend to be very difficult for LLMs to be correct on.
And obviously issues like the Entscheidungsproblem don't magically go away because we have a tool like ML that is far more computationally efficient than brute force, but still insufficient.
As LLM's will confidently present wrong answers as correct, how is that helpful for students?
> Feed forward Neural Networks, like LLMs using attention are effectively Directed acyclic graphs, so it has little it can do to 'evaluate from the outside'
I thought you were talking about the math that's being taught.
No, you can't prove the tutor is mathematically consistent. Is that supposed to be a problem? No tutor in the history of the world has ever met that standard.
No matter what you think of the current state of LLMs, that bar is so high it's meaningless.
> stressful for kids and disengage from the subject
In my limited experience, stress and disengagement are almost always the main culprit. Once someone decides, "I'm not good at math", it becomes a lost battle that they'll pretty much never revisit or seek to relitigate.
In my family growing up, we were all taught Algebra as early as 4th grade by our mom because the Philadelphia Catholic School system taught some absurd system where everything a simple education in Algebra can solve is replaced with a massive blackboard-covering table of rules to memorize that I wish I had a picture of because nobody believes me. Anyway, knowing Algebra helped all of my siblings get far enough ahead of the class that most everyone else disengaged while we excelled and younger siblings were frequently helped to advance even better because they had a house packed with people that liked math.
There's only so much you can blame the pandemic for. You can say that you weren't given proper instruction, but that excuse does not go very far when you have the internet available to you. I've seen the resources available, and even the most popular are good enough to get you through any 'college' level course. You hardly even need to be an expert at evaluating quality to get there either.
But really, universities should grow a spine and reject students who do not meet their supposed 'high bar'.
I do find it really strange for someone to ace "honors physics" to then fail qualifications for intro calculus. Seems that credit is highly suspect.
> really strange for someone to ace "honors physics" to then fail qualifications for intro calculus
Famously, there can be "plug in the numbers" physics. Zero conceptual understanding required, And then all you need to know about fractions is to "divide" that entry on the calculator.
For example, in a PSSC high-school physics course, I remember adding four or five terms when analyzing a calorimetry experiment, with no awareness of adding compatible (same units) energy-related terms.
In some ways a high-school student might be better served in a conceptual physics course, if competently taught.
> For Jessica Babcock, a Temple University math professor, the magnitude of the problem hit home last year as she graded quizzes in her intermediate algebra class, the lowest option for STEM majors. The quiz, a softball at the start of the fall semester, asked students to subtract eight from negative six.
Negative numbers are normally taught somewhere between fourth and sixth grade, so this is a pre-pandemic failure.
I did math tutoring at university, almost two decades ago now. At the time, there were three semesters of remedial math available, plus the tutoring program. The state (California) eventually got mad and made them scrap some of it, as the high schools were supposed to be covering it. Which is true enough, but obviously that wasn't happening.
I'm sure Covid made the problem worse, but not being ready for university math is clearly a decades old problem now.
This is a reminder that algebra based college physics is just kinda of trust me, this is the equation that works. This was the case before pandemic. For some reasons colleges decided that they don't want to have a calculus as requirement (usually for medical/bio fields) but they need some physics education so they came up with this invention.
Now this, I don't think this will end well on a generation level. Although it is now easier to close this knowledge gap on individuals level as they have access to a lot of online free content that can easily help with that.
Putting the math and physics in two separate sequential semesters was a problem I recognized.
I found it less efficient that the math was taught in a vacuum, and then later the application of it. For example, the math for acceleration was taught before, and separate from the real world application. When that application was taught, the math became a lot clearer to me.
In an era where schools seem more interested in passing students as a KPI, and less about actual education, I imagine the problem is worse.
> 1. Algebra-based "just apply the formula" d=vt, y=v_0*t+gt^2/2.
Referring to an algebra-based physics class?
Now also move the notion "include the units and they must work out" back to intro algebra classes, and kids could have a superpower in solving word problems.
I don't think this is really new. In first year of uni, my linear algebra professor would sometimes rant about how bad the business math students were at the subject, and they at least were in a major that required math. Students in majors that don't require math at all aren't going to have good math skills.
My calculus teacher was a business major who went back to college to get a math PhD and be talked shit about the business major math classes all the time.
I think complaining about business majors is a math professor tradition.
Yeah I know from my tutoring experience that first-year students need a lot of help reviewing the math material, or sometimes outright learning for the first time. It's a bit problem for anyone who crammed before the exam, and then didn't need to use the knowledge later on. I used to spend at least one or two sessions to review high school math before we can start on university-level topics like calculus and physics.
The way math teaching is arranged in most US public schools eschews the principles of mastery learning.
Passing a class requires a D or whatever overall, and may not even require passing a final exam.
If you 'pass' Algebra I with a C, and then you move into Algebra II, what is the most likely outcome? Maybe you scrape by with a D?
And the standards for graduating high school are low and generally not based on measuring learning or capability. That's why, at least in California, the % of students who graduate high school (complete 12th grade) is much higher than the % of 11th grade students who meet state standards in Math and English Language Arts.
The minimum standard required to graduate public high school in the US is abysmally low. You can probably get through some schools without ever showing up because the teachers and administrators don’t want to deal with you. They are incentivized to pass you even if you shouldn’t be because of how the metrics look and because a lot of underperforming students are disruptive.
However, I would have expected low SAT/ACT scores to prevent college admission if you were unable to add fractions.
I admit math is very easy for me, but it blows my mind that people don't understand algebra at least.