College students are struggling with basic math
apnews.com
apnews.com
I admit math is very easy for me, but it blows my mind that people don't understand algebra at least.
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.
I'm from German too. We use algebra as the generic term for math more or less.
I wouldn't have assumed algebra would only mean fields and rings.
The Wikipedia article algebra also links to algebra over a field.
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.
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?
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)
Is there any reason why this could not work for mathematics too ?
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.
That's not what it means. You can evaluate systems, you just have to do it from outside.
> how will pattern finding do so.
What are you talking about? We're not asking the computer to invent its own number system.
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.
Here is one paper that explains this. https://arxiv.org/abs/2301.09723
"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?
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.
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.
What age(s) is this? (We don't have 'grades' in Blighty)
The struggling student interviewed in the article lives on the east coast!
An American who cannot do basic math will often proclaim this as if it is something to be proud of.
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.
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.
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.
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.
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.
2. Calc I, where the previous formulas now made complete sense and served as mnemonics.
3. Calc-based stats, calc-based physics.
I'd recommend this to anyone.
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 think complaining about business majors is a math professor tradition.
If anyone is interested in a condensed review of high school math, you can check out my "green book" which is just a few hundred pages, but has all the essentials of high school math, see https://noBSmath.com and the PDF preview here: https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
Here is also a concept map that shows an overview of all the topics: https://minireference.com/static/conceptmaps/math_concepts.p...
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.
However, I would have expected low SAT/ACT scores to prevent college admission if you were unable to add fractions.
glad to hear there are attempts at addressing the problem.