High school math doesn't prepare most students for their college majors

For an example, the replication crisis in the social sciences. These studies, containing flawed statistics, are carried out by social scientists with PhDs who have been required to take courses in statistics designed for their discipline.

If they can’t get the statistics right, how can we expect it from teachers with far less education?

A solid foundation in linear algebra with proofs (instead of the horror that is high school geometry) will prepare both routes substantially better.

Everyone should know Bayes' theorem. It's not that hard to grasp, but it's a wildly powerful idea. Even if you aren't plugging-and-chugging, just use it to dissect logical fallacies.

If you are teacher, you learn pretty quickly that every time you invoke magic or "trust me it works this way" you lose students. Students are clever and have self-respect and they don't like being lied to, or when information is hidden from them. You have to present a coherent picture, and I think teaching statistics without calculus is incredibly difficult to keep coherent.

[1] I am not sure if it's possible to argue without limits that the normal distribution extends to infinity, yet has a finite area under the curve.

this is no longer the debate. the debate is whether kids should have to take math classes at all in order to graduate. And based on what i'm seeing, the answer the schools came up with is - No.

If you don't understand the concept of "integrating a function" they how can you possibly make sense of virtually any part of statistics? 90% of practical statistics can be boiled down to understanding the basic algebra of normally distributed random variables and then doing basic calculus on the result.

Statistics without calculus is the worse kind of statistics where students are taught to blindly throw tests at a problem without having a clue as to why they are doing this. Statistics without understanding is worse that no statistics at all.

But when I got the about two-tailed tests and p-values, it was all just so opaque. I didn't feel it was that useful, plus the whole argument about rejecting or not rejecting the null hypothesis felt too philosophical and contrived. And why 95% confidence?

I hated statistics throughout undergrad because of those philosophical contortions, which seemed very arbitrary to me.

It wasn't until grad school when I discovered the statistical learning side of things (PCA/PLS) that statistics became exciting to me -- because suddenly statistics became useful and able to predict things.

I'm still convinced the 80% of people don't need to know p-values and null hypotheses. Some might say, but what! These are used all over the social sciences! Umm no.

Let the people who need to learn that learn, and actually teach statistical learning (linear regression, logistic regression, Bayesian statistics) to the rest of us in school.

You can read the standards here, everyone learns this before high school:

https://www.thecorestandards.org/Math/Content/7/SP/

And the same topics are taught everywhere in the world, it isn't like we spend 8 years just teaching arithmetics.

This is true, and not just for high school but for college. Neither are out to teach specific vocational skills - rather they provide the underlying tools that students need to learn their vocation later on.

Sure we could stream kids from grade 1 for their "allocated profession", teaching them only with regard to their eventual job, but there's a reason we don't go that pretty dystopian route.

With that in mind we can argue that one curriculum is better than another for a specific path or other. But really that's a fruitless, and pointless argument. Future STEM students need Calculus, future programmers need Logic, future artists need Color Theory.

If I had to argue for a curriculum change I'd lobby for things like budgeting and basic accounting - but that's just me and everyone will have a different opinion.

For people who know they don't want to go on the STEM track by then already, the change is a net positive. However, it might also mean STEM loses a few more people from underrepresented groups as a result, who would have done well in STEM but are now locked out of that track earlier.

And I observe that even from friends that are reasonably well educated but have done little to no statistics.

This lack of basic stats skills does lead to bad public policies.

As it is now, you have ~3 groups of students in most K-12 math classes. A group that is bored because they have mastered the concepts being presented, a group that is benefitting from the concepts being presented and a group that doesn't have the framework to benefit from the concepts being presented.

There are of course lots of teachers that will be doing what they can to address the gaps, but it needs to be systematic.

If we taught statistics the way we teach mathematics, the entire curriculum would consist of deploying Excel functions.

The math used in jobs are also to “get things done” rather than for the sake of their beauty.

A very tiny fraction of population will ever be pure mathematicians. Most just need to know enough to “get things done”.

Considering how much society throws money at STEM, rather than art and music, it makes sense society wants a return in their investment in terms of training practical people.

This is more or less what I've found as an adult who is trying to improve their math skills. Like 90% of what I learned in school was never really applied, I couldn't tell you why I learned them, just that I was told they would be essential for my later life if I chose certain paths.

That said, as Temporal said, even this isn't necessarily done well. It's entirely possible that teaching students to derive the techniques from first principles based on actually understanding math might lead to better outcomes. But it would also be quite difficult to do and I'm not sure existing K-12 teachers are even qualified to do it. To Lockhart's credit, he walked the walk and actually taught children at a primary school. How many similarly qualified people are willing to do that?

Physics class helped me understand differential calculus because I was able to create problems that I could then solve with tools learned in math.

There are also some mathematical tools that or often quite simple but tremendously help with solving equations. Extending a formula with neutral or inverse elements for example. This is the same in almost all fields in math, but it is rarely though as a set of tools.

That said, if studies like PISA are to be believed, the problems are much more fundamental.

I'm skeptical that the kids who struggle with memorization will suddenly be able to write proofs with ease. I'd argue proof-writing is much harder to wrap one's head around than being forced to remember when to apply a2+b2=c2.

It's likely that the true ethical education children receive is observing the behavior of people in their family and community from a very young age to learn what is or isn't acceptable. Schools can't do much about that, and if anything most modern schools are harmful in this regard because they generally award petty cheating and excuse spinning. And so too do schools teach the wrong approach to critical thinking and cognitive biases. Schools reward students for conforming to what their teachers say and give students trouble when they think on their own. This is particularly true during the youngest grades. Some high-concept courses about critical thinking in highschool won't undo the damage; by that time students already learned to either conform with authority or become equally unthinking reflexively contrarian rebels.

There's confusion over the correlation of education vs ability to make good decisions -- I don't think the causal relationship between learning maths/logic and better decision making is established at all. It could be just that better "general intelligence" enables one to get good grades and also make better decisions, instead of the education having a meaningful impact.

And then there's an oversell of how useful some concepts are. I think it's fair to say that for me personally, I've had more success integrating into my life what I learned about constitutional law than linear algebra. Before this generative AI thing I don't think I've ever used applied any knowledge of linear algebra.

There's a trend of people over-estimating the usefulness of their subject. Mathematicians tend to think everyone else needs to know advanced math. Historians think everyone should learn history. Tech people here think everyone should be more technologically literate. Judges think everyone is supposed to know the law ("ignorantia juris non excusat").

In the end the body of knowledge out there is just too vast, life is too short, and it's actually a good thing that people learn different things, even at expense of being "illiterate" at some subjects. I think the "80-90%" who never integrated the stuff they were taught at school is evidence that they should have the option to learn something else instead of being forced to sit in classes that they don't feel like taking.

While in general I don't condone anti-intellectualism in the sense of being proud of being ignorant, I think to some extent it is a reaction against forms of out-of-date intellectual-elitism, especially the kind that considers people who took a classical education as superior than those who have not. For example, knowing how to build a house is definitely more useful than knowing the cause of the fall of the Roman Empire, but the house builder is presumably not looking down upon the history major for lack of house-building knowledge. But somehow there is (or at least was) a snobbishness among the educated class that did view the house builder as less "sophisticated". (Of course we all know history majors can end up worse off financially than blue collar workers now, but these days it's the STEM people who still somewhat maintain this elitist attitude and clinging onto century-old math curricula.)

And some people find it hard to accept that the classical education isn't as useful as they claim to be.

Yeah, I find that difficult to imagine too. The assumption behind that seems to be that with the proper statistical knowledge, people are able to understand ... scientific articles? Because I don't really know where else you could find a relation between cognitive bias and knowledge of statistics.

But first, articles have the statistics done. Second, knowing statistics isn't going to make you understand the article, nor spot the errors. Third, most articles still rely on poor statistics, because many of the authors and reviewers still think that e.g. null-hypothesis testing is just fine. As a corollary, most articles are wrong, and should not be used for decision making.

Kind of surprised to see that, my experience with college calc was it was so heavy on memorization, that was my struggle.

i ended up failing out of school too, lol. i was clicking around khan academy recently through their calc stuff and was surprised i struggled so much with it, 15-ish years ago.

I found that I was far more ready than many of my fellow students for math in college (though this could also be due to aptitude given my handle, even the physics and chemistry in my high school went further than AP curriculum). I attribute this to MathCounts (sadly just a 7th/8th grade thing) and the accelerated math track a friend and I were allowed to take. This track ended up getting us calculus concurrently with physics which made a lot of things there make way more sense and gave examples of real use of calculus outside of the math classroom. I also ended up taking statistics which I found more interesting/useful than calculus at the time (possibly because my intuitions weren't as applicable).

Even those who grew up with computers might not have realized you could solve problems easily with computers instead of the older methods. (I believe most people don't know they could plug a formula into Wolfram Alpha and get its integral for free.) Even if they knew, some of them might take the dogmatic view that the extra effort in learning so called "fundamentals" would build character of sorts.

So the few people who actually believe that the curriculum should be changed have to wait until they get older, rise to positions of authority, and then they maybe they could propose changing some things around. So you get at least a generation or two of students learning things the old way, because society moves slower than technology does.

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While we're on the topic -- I've always thought that the philosophy department took a long break during the mid-20th century. The logic department teaches Gödel, maybe passingly mentions Turing's work, but while everyone is busy trying to correct other people's interpretation of Gödel, they seem to be oblivious to almost 100 years of advances in our understanding of computation and programming language theory (the language of logic is analogous to programming languages). It seems somewhat similar for physics too -- interpretation of quantum physics seems to be a niche thing.

I'm not saying there's nobody who knows these stuff, but it just seems like the teaching of philosophy has stagnated for almost a century, given the weight put on learning the classics vs the newer discoveries in the past 100 years.

The nice thing is that we have the Internet these days, so at least people who are interested can find what the material they want to learn.

I took engineering myself, the Physics, Maths, Chemistry 110 "for Engineers" variations were shallow compared to the Math 100 etc core courses for those who wanted to study Physics, Mathematics, specialise in Chemistry, etc.

Mathematics, medical, and biology students who take statistics for epidemiology and other sensitive applications get a much better grounding in the pitfalls and meaningful operations of low dimensional summaries of high dimensional data.

Calculus is irrelevant for the majority of students.

That's an exaggeration. As of 2018, 65% of public schools in the US offer Calculus:

https://nces.ed.gov/programs/digest/d21/tables/dt21_225.72.a...

https://archive.org/details/AMathematiciansLament

You've put into words something I wish I could tell everybody. The impact of achieving a level of rigorous logical thinking through math classes goes far beyond "Will I use it in my job," it makes a fundamental difference to the way you see the world and think through every decision in life.

What's so unfair is that, in my experience, whether a given student in average circumstances successfully attains this in either high school or college is, essentially, a crap shoot depending on the combination of teacher, student motivation and support, and any number of other environmental factors.

Unfortunately, that seems to apply to a lot of high school math, including a lot of calculus classes. And, to shoot into my own ranks, college math and especially statistics classes are far from immune to this as well.

Nah, I would rather argue that discrete math is more valuable to students in both abstract thinking and in real life. In basic calculus, mostly you are just remembering formulas and equations like d/dx(e^x)=e^x. Not even touching the edge of abstract thinking.

If that's what you think, your math isn't advanced enough :P (disclosure: neither is mine)

These days mathematical proofs tend to be dozens of pages long and require multiple experts days to check and validate.

I'm pretty sure programming fits your descriptions of nowhere to hide flawed thinking though.

> To the extent algebra and calculus are not useful outside of STEM degrees or in “real life” is indeed what makes them useful since it allows them to remain essentially the only abstract thinking course high school students will be exposed to.

Sure, but if "useless" things are desired, why don't we teach them comparative history of Hobbit society instead?

You mention that as if it was a trivial operation, but it only works because subtracting a constant is total and injective. In general if a=b then f(a)=f(b) [the substitution property for equality] for any total function f - but the converse is not true in general.

The example I always drop is my ap calculus teacher whose students had an average test score of a 4.7, but the ap physics teacher didn't have a single student get a 5. They never shared their average for obvious reasons, but I'd assume it was closer to the national average of 2.3 based on what I heard.

Clearly the school had students capable of putting in the effort, if the teacher was capable of teaching the material.

When I realized the second part, I went to the instructor and asked: "Without the trig functions, how are they supposed to model and analyze oscillatory phenomena in business settings?".

The instructor turned and gave me the biggest smirk I had ever seen.

You are taught this is 6th or 7th grade. The students just don't remember, or lack the ability to contextualize it in the workplace.

I’d add that those are 3rd grade “math” concepts - not replacements for actual math courses.

At a more philosophical level, should schools even teach personal finance? The math itself is very simple and many adults can learn it on their own, without guidance from any instructor. All they need is a couple of good blogs and YouTube videos.

The classes schools teach and the curriculums they offer are limited. There’s only so much time in the day to teach a wide variety of topics.

So, should educators spend time teaching subjects students could otherwise learn themselves? Without an instructor, assignment feedback, etc, would people still pick up something like physics or calculus on their own?

My personal opinion is that K-12 and college should focus on complex topics like algebra, chemistry, and physics, while simpler courses like personal finance could be taught by parents or by a students self. Happy to hear the debate around this though, it’s always fairly interesting.

What way? The classic way to learn is to be proficient at multiplication up to 10x10 and then solve for the reverse.

The one noticeable difference was that my kids did virtually no proofs. When I was in school, my district used a "new math" curriculum that introduced sets in first grade, and included derivations and proofs. (I'd call a derivation a lightweight proof). High school geometry was almost 100% proofs, and it was a class that a lot of students remembered as their favorite.

My kids: No proofs. They solved lots of problems where they were expected to inspect a problem, choose an algorithm, then crunch through it to an answer.

Proofs were what made math come alive for me. I could crunch numbers in science class, or on a computer, as I learned programming in 11th grade. I ended up majoring in math, and then added a physics major as well.

Math is an extremely confusing and fraught topic for parents, because we all viscerally know it's important for some reason, but nobody can really put their finger on why. Some parents treat it as a form of obedience training, or expect that it will magically confer special thinking skills. We know math is a sorting hat for getting into vaunted STEM programs in college.

Very few people use their school math after they finish school. In school, it's treated as a tournament, to reach "levels" and get good "scores." Many of the brightest kids are repelled by this. I would have been.

The college math topics are the same as they've been for 50 years. A number of years ago, in between jobs, I taught an introductory math course at a Big Ten university, and the students didn't touch a computer for the entire course. My office didn't have a network connection.

If it were up to me, I'd add a lot more computation and data work to K-12 math. And I'd bring back proofs. I envision a balance of four quadrants, not in any particular sequence, but perhaps in a cycle:

1. Arithmetic, which is symbol manipulation up through calculus

2. Computation, which is using computers to solve problems

3. Working with data

4. "Theory" which I associate with proofs and abstract topics

My advice as a parent is, first, be prepared for them to follow their own interests. This might include not being interested in math. Be prepared to help them deal with the competition, and to not let it cause them to lose interest. Next, treat it as something interesting and fun at home, separate from grinding through problems.

Once your kids get into higher grades, you may end up seeing them be taught the "tricks" that you may have come up with yourself as a math person in previous years. These solutions leverage the students ability to have a functional mental model of how the math works, even if they haven't done as much rote memorization. In a modern world with calculators everywhere, it's better to teach students to be able to identify incorrect calculations quickly over repetitive practice on large, complex sums and products. (Times tables up to 12x12 are still ubiquitous and emphasized, as having those down helps a lot when doing other math.)

this is good overall, although of course it's possible to ruin anything via bad teaching.

Where do you find this shared view of what is considered the least valuable part of the curriculum?

Yep, this holds true for programming too. You either have the mental capability to turn a problem into a set of instructions that can be coded (and then grind through the syntax and language specifics) or you don't. I'm not that young anymore, so my first formal programming course was in college (at the time when pretty much every stem student had a computer at home, some even a laptop for college), but it was just like this... with zero correlation to math grades, some would understand the concept and break the "idea" down to basics programmable steps, and some would get stuck at the "design" phase.

Also, this was basic programming... eg. fibonacci sequence, so they could explain recursion after that. Basically, some would understand "ok, I have to start at zero and go on, but i don't need to remember just the last two numbers, so two variables, previous and previousprevious, and a loop,..." (and then debug the for loop since it starts at zero, and <= was used for n or whatever), and some would get stuck in a mental loop, if they have a "n", how do they get the previous two numbers, if they need previous numbers for those too.

So this is way before the level of patterns and generalizations, where you need the subject matter knowledge to even know, if you should fit the curve to a straight line or some higher function.

As for your use case, sometimes, some people are hired to not understand things.

Can you share that proof?