Against Algebra
theatlantic.com
theatlantic.com
Quoting from the OP:
> The traditional arithmetic I learned in the early grades made sense to me because I could relate it back to real-world things. Later, when I was learning how to design, finding the area of a circle proved essential to practical tasks such as sizing hydraulic and pneumatic cylinders. I could do trigonometry by visualizing, for example, the cables on a suspension bridge. But algebra: no.
For a different perspective, I recommend reading instead Susan Rigetti (nee Fowler)'s posts about overcoming challenges to learn Math and Physics. She says she had to solve what felt like "millions of exercises" to become familiar with many abstract concepts, until they finally clicked for her and she was able to reason about them. In her words: "Solving problems is the only way to understand mathematics. There's no way around it."
https://www.susanrigetti.com/math
https://www.susanrigetti.com/physics
https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgli...
I think that numbers create all sorts of problems because people focus on them as they are familiar and miss the bigger picture.
Abstract algebra (i.e. actual algebra, as opposed to doing useless stuff to poor polynomials) requires one to have developed the mathematical maturity to iron out informal thinking.
Good luck explaining "prime" and "irreducible" are not the same.
I think a path that would work better is through problem solving, like olympiad problems. But even that definitely isn't for everyone.
Without the proper context, group theory and abstract algebra are deeply uninteresting subjects. It would be like trying to read Bertrand Russell before you learned your ABCs, ridiculous. It's important to start with numbers because they are familiar.
This involves paying teachers paying more and treating them with respect? Nah, lets have endless conversations about how whether we should not learn math at all!
More money isn't the solution.
Have you been to the US? Because most people here don't respect teachers. And we pay teachers almost nothing.
That is largely not true, though certainly true in some cases. Teacher salaries are set by the local school district and range from "reasonable" to "not great".
Overall, teachers make about $51,000/year while bachelor's degree holders as a whole make about $59,600/year. Against that, you have to recognize that the teacher pay is for nine months, while the bachelor's degree number is for twelve months. If the typical bachelor's degree holder worked only nine months, one would expect their salary to be about $44,700, considerably less than the typical teacher.
Sources:
Or do you believe that if the profession were more respected, it would attract a better, more qualified class of teacher, who could then better educate the students?
And, to be clear, I think teachers are generally qualified. Math just represents a very acute failure mode of our education system.
That issue is highly political though. Even when setting aside the issue of selling to voters a pay raise for teachers. Good math teachers are specialists with a math education. To attract them to teaching you’re going to have to pay them more than other teachers. How do you sell that, politically, to teachers’ unions?
There’s a lot of cognitive dissonance going on with non-specialist teachers teaching math. They’re teaching bad habits and passing on their own math anxiety to students but if you confront them about it they get very defensive and start talking about school funding and textbooks and materials and computers.
The truth is: that’s all bunk. Third world countries are better at teaching math. They don’t get distracted with computers and they don’t waste time and money redesigning the curriculum and printing new textbooks all the time. They teach math in the old, straightforward style and emphasize practice.
Paying english teachers less also means you get bad english teachers that are bad at teaching critical reading (which..yikes!), so I don't really see the point of that pay discrimination.
I would absolutely advocate for removing administrative bloat, but we should be recruiting teachers like tech workers.
I haven't seen any evidence that any particular teaching method delivers tangible improvements in critical thinking skills over the school term. Do you know of any? It's important to measure skills at the beginning and end of a term (and a followup years later) in order to show improvements at the individual level. Otherwise you risk selecting for students' prior ability.
As far as I know, professions like teaching, nursing etc. don't work that way. I personally know about 3 teachers who genuinely cared about their students but could not survive on teacher salaries (2 are now real estate agents and miss their kids :/).
It seems like paying the same people more money to do the same job won't change outcomes in any meaningful way. Why would anyone expect it to? Paying teachers more in a competitive market, on the other hand, might has a dramatic impact as we might likely end up with very different teachers.
I'll read the linked documents at some point, but taking it per se, it should be noted that doing problems as a method of understanding may be necessary, but is not sufficient. I had a professor who taught how to solve problems and then assigned way way too many problems. I could manipulate the symbols, but I didn't grok what I was doing until I picked up a different text and started learning on my own. How many problems are needed definitely varies from person to person, and I'm not entirely convinced there aren't other ways to understand mathematics; but I suspect problems are the fastest and easiest.
In music I’ve heard the answer is that no, you can’t speed it up, you have to do the practice. But then how to explain the Mozarts of the world who somehow seem to have, innately, the knowledge that for others requires training and time?
Those kids didn't really start out with "innate" knowledge, or anything like that. They get intensive training starting at a very early age, that's how they do better than others. Also, practice is not necessarily boring. When you're actively learning/mastering a skill at close to optimum effectiveness, practice is quite enjoyable and it's easy to rack up a lot of it.
Why would it take everyone the same amount? It isn't like there is a natural law that you need to drill the same problem 20 times to understand it, why wouldn't someone's brain be able to process it and understand it after doing it the first time or even just looking at someone else doing it?
I found a joy with math only by accidentally reading biography of Newton, Godel Escher Bach, a book about Cantor, and a compendium of math papers for educators.
Then I could see all my reasons for avoiding math were excuses for poor insight to the beauty and interrelatedness of mathematics.
You do the work for intimate understanding for each move. If I could learn mathematics like learning sports (simple ideas worked into muscle memory) I would appreciate the result, and you can keep the abstract problems. I’d rather solve real problems.
Isn't the opposite true? I thought the 'learning styles' theory had been debunked by science?
https://www.psychologicalscience.org/news/releases/learning-...
A few other, major issues:
* Math has a severe stigma. How many people do you know would readily confess "I'm just not good at reading," almost as a badge of pride? This is common with math.
* Most teachers don't have a strong math foundation. They are not acquainted with some of the foundational ideas that can really enhance teaching and learning math. In some cases, teachers who don't like math will perpetuate the above issue.
* There is little incentive for those strong in math to become teachers. Why go teach introductory math when you can become a software developer and make 4x the salary, with less stress to boot?
I'd look to remedies like paying teachers better, improving their workload, and even revising math curriculums to focus on concepts over testing before jettisoning a whole field like this article advocates.
Perhaps because it is genuinely less important on the pyramid of human skills. Reading, language comprehension, and critical thinking are more universally needed than, say, matrix multiplication or integrals.
2. I may be biased, but I believe math is distilled critical thinking. The difficulty is teaching math critical-thinking-first as opposed to test-first.
I actually think this is likely not true.
If you were to take an 18 year old who can't read, and another one who can't do addition, which do you think is less employable?
There's a lot of jobs one can do without any math. Almost none one can do without any reading.
There's more to it than all this of course, but I think literacy is the clear winner compared to numeracy.
The person who can't read can be a traffic guard, server (with the right cash register), bricklayer, or any of a number of jobs. However, all of those people need to know that they worked X hours @ $15/hour and should be paid 15X. Otherwise they will never know if they were ripped off or be able to plan for the future.
That said, addition is a very low bar--I cannot think of a single job that would not require at least basic addition.
Based on current wisdom. Current wisdom has a spotty track record though. It was once an obvious truth that reading was only for a select group. Universal literacy would have been as absurd back then as the notion of universal math literacy is now. Which is depressing, because one would hope that at least this historical analogy, and it’s implications, had occurred to more people.
> "We don’t need Americans to be better at algebra, per se. We need future generations that can build and repair infrastructure, overhaul energy and agriculture, develop robotics and AI."
If the issue is how algebra is taught (dissociated from physical objects), well, perhaps there's a curriculum problem. The fundamental concepts of algebra - such as commutation, association, and distribution, do indeed have nice visual and physical representations:
https://www.mathsisfun.com/associative-commutative-distribut...
Everything else in algebra, trigonometry, even into calculus and more abstract math relies on understanding how those rules can be used to manipulate equations, and then you can put the equation on a computer and build models of bridges, robots, power grids, etc. You can also grasp compound interest and so not be screwed over by shady mortgage brokers pushing adjustable rate loans... (although one wonders if such widespread financial literacy is really something the oligarchs who control both political parties really want to encourage).
Perhaps the real problem is the failure of the America system to place the appropriate value on education and hence invest resources in educational programs, and that's the real reason for the poor performance of American children on math relative to all other industrialized countries?
Imagine an article saying we need less English and less composition in college and high school, except you wouldn't see such an article because writers do not hate writing (at least not beyond some ironic sense). But sure, if you ask any teenage with a 10 page report due next week, they definitely would like less writing in their curriculum. I feel like we will however never seen op-eds critique writing and generally humanities in education because those who write op-eds value it. They however, do not value mathematics in their own lives (probably having others do the mathematics they would otherwise need to do for themselves), and thus they incessantly attack it.
I hate to make a meta-psycho-point but I must:
Some part of me wonders if stuff like this is some people in the humanities having a reaction to the status quo today in academia where they are undervalued--I do feel that in status and funding in education today, the humanities generally does not receive the respect they deserve. Too much emphasis is given to STEM (and lately, STEAM which still leaves the humanities out). That said, I don't think attacking STEM is the way to ameliorate the issue, and it really leaves me with the same feeling I had when my peers cracked jokes about "the English department," that they simply did not understand what they study and are painting other fields and experts with broad generalizations.
On the contrary, I've seen many such articles. Education in general has been under attack for quite some time now.
I thought the “visual learner” classification was a myth.
However I would caution not throwing the baby out with the bathwater. There is a growing body of evidence that some people are better or worse at visualizing, have a stronger or weaker internal monologue, etc. than others. There are reported neurophysiological differences as well as task behavior differences if you look at people with Aphantasia for example.
As the other article sent to you mentions, someone who is not very visual could potentially benefit even more from visual examples, so these results are not really in conflict with the failures of education theory. It's also not clear how common large disparities in the modalities of thinking are, so perhaps it is not something worthwhile to consider for mass education policy regardless. But I think in a neuroscience context it remains an interesting topic for future research.
I'll also note that the author of the original piece is autistic, and she has clearly demonstrated savant-level abilities on certain visualization tasks. It's not surprising that she would feel this way about "learning styles", because her entire life experience has likely been extremely lopsided towards visual thinking.
That may be over-generalizing. From what I've seen, the studies that debunked this idea debunked something fairly specific, not necessarily every possible formulation of what this phrase could mean to people.
If a particular subject doesn't "click" for her, it will be a lot more of a barrier than it is for most. But, conversely, how many more like her have been accidentally excluded by cognitive barriers that most of us can't even see?
What she has learned is that things that can seem impossible can sometimes be broken apart into small pieces and conquered with hard work. The author probably had certain subjects come easy to her. When it got hard she probably didnt know how to do the repetitions required to become good. Instead of becoming good, she is trying to tell us in STEM disciplines that the subject simply isnt necessary.
Algebra is fundamental to all other math I dont see how you can do statistics (as the author mentions) without algebra. In fact probability is much harder than algebra and probability problems often times require algebra. Further the work habits you learn to master algebra are used for other abstract subjects.
As you get into the sciences and advanced statistics, you might need calculus. As you get into engineering you surely need calculus. There are often times versions of the subjects that allow you to memorize the equations and the author probably doesnt realize the difference between memorizing the statistical equations to be able to execute them and deeply understanding how they were derived. It is the difference between an electrician and an electrical engineer.
As a biology student you might not need much math, but as a scientist in general you do need physics and chemistry. Without algebra, how can you do even basic newtonian physics?
I have an electrical engineering and a microbiology degree. The biological sciences require little to no abstract problem solving, they are essentially all memorization. It is easy to get by without having to do a certain kind of thinking that exists within algebra. That kind of thinking though exists in virtually every engineering class, physics, and even a little bit in chemistry.
The author is essentially implying that animal sciences is stem and animal sciences dont need math, therefore STEM doesnt really need math.
There is simply no evidence for it, and study after study shows that trying to match method to perceived learning style does not improve performance.
It is one of those things that seems like it should be true, and therefore people like it, but there is no evidence that it is in fact, true, and plenty against, for well over a decade at this point.
See:
https://www.frontiersin.org/articles/10.3389/fpsyg.2020.0016...
http://doi.org/10.1016/j.compedu.2016.12.006
https://www.educationnext.org/stubborn-myth-learning-styles-...
https://www.psychologicalscience.org/news/releases/learning-...
etc
Everybody should read "Thinking the Way Animals Do: Unique insights from a person with a singular understanding." https://web.archive.org/web/20210417083222/https://www.grand...
I think she is right.
Everybody should study as much math as possible in school, more than today. But nobody should study math abstractly just to pass exams. They should study more but only at the rate they understand, in a way that they understand.
If everyone would study math 5h/week for 9 years with speed they understand and grok what they learn, some would learn to master only 50% of the math curriculum. Some would master 200%, some even more. But even those who would master only 50% would have better math skills than those who pass all exams but don't undestand.
Why does it have to be one or the other? Besides, algebra can also be taught in a more visual way and featuring more concrete reasoning, such as via complex word-problems that can be solved with a mix of arithmetic and algebraic approaches.
The point of the article is that it should not be one or the other, different students need different things.
It is the removal of tracking, the removal of vocational education, the denial of differences in students and the insistence in homogenized education that is wrecking schools.
If only 37 percent of 12th graders can do the math needed to start college, maybe only 37 percent of 12 graders should be going into debt for a college education.
Maybe you wouldn't if all teaching was appropriately structured. But remedial instruction is where the flaws of the nowadays-conventional approach of expecting students to "inquire" about math on their own and come up with their own ways of problem-solving are most evident.
Wouldn't that mean tracking is a better solution? Let the advanced kids who can quickly understand/infer the ideas and techniques move on, and walk through the details/give lots of practice to the other kids?
I am 100% confident that I would not have gone into engineering if algebra had not been part of my compulsory education. And I'm very happy with the career I discovered for myself. Neither one of my parents were engineers. And I wasn't a particularly good student in middle or high school. Therefore, I doubt anyone in my family would have thought to enroll me in extra curricular math. I discovered that I was good at those things late in high school.
So I see advice like this as worsening class divides and increasing the likelihood that you live the same life as your parents.
This ignores that much of the fundamentals (like Math) are less about teaching you the basics but teaching you the skills you need to learn as an adult. At least this is how I have come to view my time in school as an adult.
Now obviously there are exceptions of this (I mean you did have to be taught to learn and to read) but with this basic fundamentals when you are presented with a problem later in life you have those skills to pull from. Even of the algebra is not exactly what you did, The idea of "solving for x" should be so ingrained in your head that when you need to solve a simple problem that is basically exactly that, you do it without even realizing that is actually what you are doing.
Even as a software engineer, most of what I learned in school I am not actually truly using on a day to day basis. Except when I am working on learning something new.
Maybe we could get rid of "algebra" but what exactly would it be replaced with? Something that is algebra in all but the name?
Also my system had different levels of classes, I know some kids were taking pre-cal in high school while some only finished algebra 1 (2?, I don't remember)
It isn't uncommon for parents to say how much they hate math so of course their kids are going to start to think that.
Or to talk about how much they hate school.
Leaning should feel rewarding and yet, it feels like it isn't in this country.
It seems myopic to search for the answer to this discrepancy purely within teaching styles. In high school, coincident with the onset of puberty, many students simply become more interested in achieving social goals than academic learning.
The author makes this ridiculous assertion. When people first do anything they are always bad. Over time, through an accumulation of small efforts they can become good at something.
Asking what are you good at is the most useless question.
Depending on what you mean by "bad", this is not necessarily true. Some people are better at certain things than others if you control for the amount of practice they've had. Take a group of children who have never played the piano. Give them a recording of a simple song, and tell them to listen to the song and play around with the piano for 1 hour and try to play something that sounds like that song at the end of that hour. Even with zero practice, some people will do better at this than others.
If you wants things to be even more visual you could use Penrose diagrams [0], though I don't find them particularly intuitive.
[0]:https://en.m.wikipedia.org/wiki/Penrose_graphical_notation
I disagree. If “a + b + c” suggests “+” is associative, and “a × b × c” that “×” is associative, what about “a - b - c”, “a / b / c” and “a ^ b ^ c”?
Or are you suggesting Σaᵢ and Πaᵢ suggest associativity? If so, how?
(I also think you mean “commutativity”, not “symmetry” (https://en.wikipedia.org/wiki/Commutative_property) symmetry typically is about relations (https://en.wikipedia.org/wiki/Symmetric_relation)
Chesterton's Fence applies to many of the complaints; the OP doesn't understand math, but implies it's a bad thing they would have been prevented from becoming an engineer because of that. Which is just wrong.
https://www.newyorker.com/science/elements/california-studen...
- The COVID pandemic forced students' and teachers' experiences into the open; schools still haven't returned to "normal" since 2020
- Folks here are trying to grapple with the roles technology can play in learning and in our education system
- Software engineering and related fields require a decent education; perhaps folks are worried about who may be prepared or under-prepared for the field in the future
> Students need more exposure
Also The Atlantic:
> Paywall