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

143 points by bikenaga 3y ago

grepfru_it 3y ago
I have a kid starting 1st grade and I have just been exposed to common core math for the first time. At a glance I do not understand how it prepares for a future in mathematics or related fields. Interested in hearing from others who have been through common core math first hand or raised a child through the program Also the way i learned to divide in my head is completely backwards and, though it works for me, when i tried to explain it to my wife i realized how completely backwards and inefficient it is, so maybe common core helps with these and other hard-to-internalize concepts?
- rahimnathwani 3y ago
  When you say 'common core math' are you referring to the standards themselves, or a particular set of textbooks that cover the required material?
- koolba 3y ago
  > Also the way i learned to divide in my head is completely backwards and, though it works for me, when i tried to explain it to my wife i realized how completely backwards and inefficient it is, so maybe common core helps with these and other hard-to-internalize concepts? What way? The classic way to learn is to be proficient at multiplication up to 10x10 and then solve for the reverse.
  - deleted 3y ago
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- irrational 3y ago
  Well, one thing is that I can’t help my kids with their homework. I simply don’t understand the way they do the math these days. They get the same answers, but go through entirely different steps. I remember one parent teacher conference where my kid was in attendance (4th grade). The teacher was showing my kid how to do a problem. I was able to get the answer in my head, but she was using all these boxes and things to get the answer and I could not follow along. I still have no idea how she got the right answer doing that. It seemed incredibly complicated.
  - practicemaths 3y ago
    I do not have a child or direct experience with common core. Personally I suspect issues lie more in teachers themselves not being taught correctly than the actual concept behind common core though. From what i can gather it is trying to introduce the concept of factoring earlier which in my opinion makes sense since that is really useful in Algebra and above.
    - j45 3y ago
      Learning to teach is very different than learning, let alone first principles learning for children. Having basic understanding, then skills, then competency to remember and test via assessment in that order can be missing even for graduates. Universities focus on teaching young adults how to teach children. The rest of the world as adults don’t learn how children do.
      - practicemaths 3y ago
        And many teachers have continual education. That means the implementation of common core is also upon teachers with decade(s) of experience being told to teach differently. Regardless, if your child has a nack for maths that is a great advantage over the other children and worth investing in since so few people bother to learn it at all. Additionally it leads to better reasoning and logic in other academic and professional settings as well. Which means a better chance at a higher income and ability to sustain and secure their livelihood and thus afford having their own children possibly. Which means a better chance for continuation of a little bit of yourself.
  - monocasa 3y ago
    The point of most of the common core math changes were to teach numeracy rather than rote arithmetic. Remember how some teachers said "you won't just have a calculator on you at all times" and it turned out that we all carry around what would have been supercomputers for the time? Early math education realized that too, and shifted the focus to quickly estimating solutions through understanding the relationships of their numbers so you can go "hey, something's not here" when you work with your calculator, rather than just accepting whatever garbage out came from your garbage in. It is complicated, but the point isn't to get the right answer per se, it's to give kids a geometric intuition of numerical relationships by going through the whole rigamarole.
- analog31 3y ago
  My kids are now college age, but when they were in K-12, I looked up common core, since the standards are published in several states. Honestly it looked quite a lot like the math that I learned in school, give or take. I took exactly the same school math subjects. 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.
  - calvinmorrison 3y ago
    for the general populace, good arithmetic, with a smattering of practical geometry and trigonometry should be the focus for more years of education.
- secabeen 3y ago
  Primary school common core math is focused on teaching numeracy and conceptual understanding. Students who want a future in mathematics or related fields will take more formal rigorous math in secondary school, often via a "compaction" program that resembles older math programs that result in completion of Calculus by the 11th or 12th grade. 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.)
  - zozbot234 3y ago
    Repetitive practice is important - it's about getting exposure to formal reasoning in a "toy" context. You can't get "conceptual understanding" in math without learning what formality and rigor is for.
- currymj 3y ago
  i think you're exactly right -- it is trying to take some of the mental tricks that "good at math" students would intuit for themselves, and teach them explicitly. this is good overall, although of course it's possible to ruin anything via bad teaching.
Scubabear68 3y ago
I want to know what world this author is from. High School math in public schools has been dumbed down repeatedly over the years. Calculus in High School? Increasingly rare to non-existent. What we need is the opposite - a return to traditional curriculums, and stop non-sense like a 5 credit course in graphic novels (a thing in our district school).
- threeseed 3y ago
  Why is that course nonsense if it's elective ? Calculus is irrelevant for the majority of students.
  - Scubabear68 3y ago
    It’s nonsense because this is a public high school, government mandated and paid for by taxes.
    - ceejayoz 3y ago
      Helping kids engage with literature they enjoy seems a lot worthier of my tax dollars than, say, teaching cursive.
      - BlackjackCF 3y ago
        Please tell me they don’t teach cursive anymore. I remember being fed the lie that I would have to use cursive as an adult. I don’t even really write anymore…
        ceejayoz 3y ago
        They do still teach cursive, at least here in New York state.
        ern 3y ago
        Cursive is making a comeback. Google it. The education establishment is often divorced from reality.
hn_throwaway_99 3y ago
This is essentially the "calculus vs. statistics" debate that has been going on for some time now. I'm definitely on the side of making statistics mandatory, and then making pre-calc/calc electives for motivated students who want to enter STEM fields. Primary reason for this is that I believe a base knowledge of statistics is so important for everyone in modern society, and it directly goes to better being able to evaluate the constant flood of information we are now all subjected to.
- deleted 3y ago
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- fasthands9 3y ago
  I agree - as long as it's not an excuse to dumb things down statistics should be taught before Calculus. It's a quirk of history, I think, that we have it the way it is. Statistics is still going to be important to understand for 99% of STEM majors, so I don't think there's much downside to flipping the order.
  - Doozy7685 3y ago
    [dead]
- cowsandmilk 3y ago
  AP Statistics has existed since 1996 and my high school (and many others) offered it as a path for students in addition to the calculus path. That is, you could head down the statistics path or the calculus path (or both) depending on your interests.
  - smeej 3y ago
    I finished the AP Calc path my junior year, but was told colleges would still expect to see a 4th year, so I signed up for AP Stat. It was mind-numbing how easy it was by comparison. We got to the end of the semester and I realized we'd only just reached some of the principles from the first week of calculus. I don't remember why he was so proud to point out that we could now calculate derivatives (it was a long time ago), but I was thoroughly unimpressed. By the second semester, I had my college acceptance letter, so I didn't bother with the other half.
kurthr 3y ago
Maybe history, social studies, government, or economics classes in highschool should use a spreadsheet to show basic data analysis and statistics (mean, median, mode, stdev). It would be good if there was a more basic understanding of how an "average" can hide a lot of inequality. I don't really see getting rid of angles and algebra to cover something "outside of STEM".
- nemo44x 3y ago
  You assume the people teaching those subjects could do that.
- starcraft2wol 3y ago
  > (mean, median, mode, stdev) 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.
  - dagw 3y ago
    This is where taking a more holistic approach to curriculum planning would pay off. If the weeks after you were taught these (or any other) concepts in math class they start showing up in your science and geography classes, you'd probably remember them better and have a better understanding of how to use them.
nebula8804 3y ago
Man oh man does this bring back past trauma. I graduated high school in 2006 and despite me being in one of the top half rated high schools in my state and having a consistent B average, I was not prepared at all for when I entered my engineering school (NJIT). A lot of my math was lacking fundamentals and instead was just memorization and 'plug and chug' style learning. That does not fly at university especially an engineering school. Schools (at least my school) really expected you to learn fundamentals as their tests would not just be a variation of the problems taught in the lectures but a unique twist on the concept itself. The school tested me on my skills before entering and put me in a 'remedial' course called Pre-Calculus with a Calculus Perspective. They even wrote a custom textbook to really fine tune getting prepped for what was to come. (Calc 1, 2, 3, Diff-Eq). I wasn't enough because the new thought process requires a great deal of effort to internalize in the timeframe given. I was barely treading water my first semester and ended up failing math and physics and got put on probation. I eventually graduated and am doing quote ok in life career wise but it was an extremely painful ordeal. Not being prepared and really ready to go remains to this day one of the biggest regrets of my life. I really feel like I let down the people who made my education possible (parents, teachers, schools). Sometimes I wonder why is there such a dumbing down of mathematics in the US? The US is powerful and at the top of their game because of science and technology. Why try your best to kill your golden goose?
- jaybrendansmith 3y ago
  Similar story here. What I learned in math in high school was an absolute joke. It in no way prepared me for physics and math in college. In addition, most college majors do not need it, they need an understanding of statistics, and a much better understanding of 'business math' and algebra. I think dividing kids into two or more tracks does not hurt them, because they can always take extra course work in college if they ultimately decide they want to pursue STEM. Where it hurts them are the standardized tests, this seems to be the issue.
- tayo42 3y ago
  > instead was just memorization and 'plug and chug' style learning. 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.
smeej 3y ago
I have a friend who has been a high school math teacher for 15 years. When she started, she asked me and the rest of our friends to make note of any times we use algebra or more advanced high school math in our daily lives. Both times it has happened, I've been sure to let her know. Once, the Pythagorean theorem helped me figure out how long the sides of square table decorations needed to be for my sister's wedding if they were to be inset in round tables with 5-foot diameter. The other time, I was able to prove, conclusively, that I could not possibly have been speeding before another driver caused an accident, but none of the police officers could follow the math, and since I wasn't a "certified accident reconstruction expert," they wouldn't believe me anyway. (Hiring one would have been more than the repair cost.) It didn't go over well when I told the supervising officer that just because he couldn't do the math, it didn't mean it couldn't be done, even by some dork who had a liberal arts degree but had, in fact, studied calculus in high school.
- j45 3y ago
  One thing that stood out in university was learning the algebra version to so many basic math formulas in any field, whether it might be physics, math, biology, chemistry, or most anything. As for your use case, sometimes, some people are hired to not understand things.
- hutzlibu 3y ago
  "The other time, I was able to prove, conclusively, that I could not possibly have been speeding before another driver caused an accident" Can you share that proof?
  - smeej 3y ago
    I don't still have it. It was years ago. I had the accident record pulled from the vehicular equivalent of my car's black box that said how fast I was going at impact. The distances involved made it such that I could not have braked to that speed from a higher speed than the speed limit in the required space.
j45 3y ago
College doesn’t often enough prepare a graduate for a career in their field of study. Math is important because it underpins so many decisions improving a tiny bit better over a long period of life, if you can measure and compare. Without it is often more profitable for those who want profit. edit: clarity
nemo44x 3y ago
How about we start with high school students being prepared for math that concerns their adult life. Like how debt works, compound interest, savings, budgeting, etc?
- smeej 3y ago
  Reminds me of that joke that crops up in April, about how grateful we are for all those hours in our high school math classes this parallelogram season.
  - nemo44x 3y ago
    Geometry and abstract math is important. But getting the practical stuff into even remedial heads is necessary to. Should do both.
- jmye 3y ago
  We had home ec in junior high and high school - students just chose not to take it. I’d add that those are 3rd grade “math” concepts - not replacements for actual math courses.
- SamuelAdams 3y ago
  Many schools do teach this. It’s not required but some schools do offer it as an elective. However, as practical as it may be, many teenagers just aren’t interested in learning about personal finance. 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.
jccalhoun 3y ago
I teach Communication at college but I was a math minor in undergrad. I hear my students complain about math class all the time, how they hate math, and it isn't rare to hear students say they have taken the same math class multiple times. I've glanced at the math book and it is basic stuff. I don't know how but we need to get kids to not hate math before we can do anything else.
- tayo42 3y ago
  I think its because of how its taught. one thing is its often for a lot of people just learning for the sake of learning once your doing anything like algebra or harder.
- deleted 3y ago
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- kwk1 3y ago
  "A Mathematician's Lament" by Paul Lockhart is a pretty good read (25 pages) on the topic, I think: https://archive.org/details/AMathematiciansLament https://archive.org/details/AMathematiciansLament
hutzlibu 3y ago
Data point, (or rant) about high school (Abitur) situation in germany: it is the same or worse. I was quite good in math in school, yet in the beginning did not understand anything math related in university. Which was no surprise, because the math in school for me was basically memorizing algorithms to solve known problems of style X. But not really understanding it. And then in the final years in school, while in theory doing higher math, the main focus shifted on how to use a graphical calculator to do the math for us. But in university I suddenly had to understand what I was doing, to be able to use it. Oh my. So it was just really hard grind, to recover all that I did not learn before (that grind was highly beneficial, though). In theory the universities knew about the problem and they complained about it to us - but as far as I know, nothing has changed since then. So Math will remain a weeding factor for lot's of students, who otherwise might have been great engineers, but maybe had the bad luck, of having bad math teachers on top of the bad curriculum. It is also a really bad sign, that math is so universally hated among students. Because math done right, is just thinking done right - and I think the world might benefit from more of it.
squirrel6 3y ago
Honestly, the bigger issue here is that good math teachers in the US are just in short supply. Sometimes I wonder if the way we teach algebra here is flawed, like something pedagogically or conceptually, and good teachers break through that to give students what they need to prepare their students.
bigger_cheese 3y ago
Even in the STEM world I don't think statistics is taught all that well. I studied Engineering a STEM subject, this was sometime ago (2000's) and at that time we were required to take a "statistics for engineers" course in the second year of the degree. The subject was mostly useless (it pretty much boiled down to "plug this data into Excel and this is how you interpret the ANOVA output"). What I really wish they focused on was regression analysis and especially non-linear regression, logistic regression etc. I had to learn all about it in the workplace after graduation. Even the statistics knowledge we were taught was incomplete - for example I did not learn about Cross-validation at university. talking to more recent graduates I don't think the situation has improved all that much in the intervening years.
- defrost 3y ago
  The "STEM World" is a large one and "XXX for Engineers" courses in undergraduate university are long on material to rote memorise and short on depth required to understand and expand. 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.
nobodyandproud 3y ago
How exactly does pre-calc really prepare a student for calculus? I remember taking Trigonometry—useful—-and then taking pre-calc.
akasakahakada 3y ago
Funny that most humanities degree holders that I am aware of do not agree that humanties events can be expressed in numbers. They can't comprehend anything rather than over simplified stories. What I mean by degree holders, I mean Phd degree holders, professors. What being said in the article reflects misunderstandings of math from the arts and humanities community. What they need is a general understanding of discrete math rather than specifically statistic. Calculus is valuable to STEM because of the smooth and continuous natural of macroscopic world. Contrast that cultural/social/econ events are discrete. If you want to draw a beautiful smooth line which fit so badly, that do not tell the whole story. However we see people drawing line cut through a circle in a graph in most humanities papers and they claim there is a positive relationship whatsoever. And here is how flawed the survey is. >>> Survey target: Majors that require calculus were excluded. >>> Many high school math topics were unimportant to college professors. For example, most professors said they wanted students to understand functions, particularly linear and exponential Linear functions are already taught in pre-calculus. You want calculus but you don't want calculus. >>> the ability for patterns and relationships and make generalizations This is unrelated to math. More related to IQ. Seems that students in the non STEM realm generally lack the ability to find patterns as professors witnessed. People should take more concerns on this phenomenon.
- ajsnigrutin 3y ago
  > This is unrelated to math. More related to IQ. Seems that students in the non STEM realm generally lack the ability to find patterns as professors witnessed. People should take more concerns on this phenomenon. 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.
dboreham 3y ago
US-centered response, assuming the article is about US education. The article is off track vs what I observed with my kids math (typical middle of the country small public hs). Nobody gets to do trig until they've done boat loads of stats. The thing I saw missing was the kind of top notch teacher that can inspire and intrigue the students. This can only be addressed by paying teachers much more. That's not going to happen. Our school only had one "real" math teacher, and he retired last year.
- presentation 3y ago
  Definitely ought to be paid more but I suspect even with more pay, most teachers don’t have that special sauce.
- ApolloFortyNine 3y ago
  Paying more goes hand and hand with testing teacher (or holding them responsible for students grades), and generally that's not very well received by teacher unions. 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.
photochemsyn 3y ago
I don't understand the statistics-vs-calculus debate at all. Probability theory is at the heart of statistics, and how can a student understand statements like 'the integral of a probability density function must equal one over its possible range' without taking some calculus? The problem with calculus as taught today in most schools is that most of the mathematical tricks for integrating functions that they hammer the students with are essentially useless in real-world problems so you might as well just start with numerical methods of approximation right at the beginning, once basic concepts like 'the integral of a function corresponds to the area under the curve of that function' are grasped. I know many math purists are dedicated to the pencil and paper approach, but it's not all that useful for non-math majors who will certainly be building or using computational models almost exclusively. Another prerequisite for statistics should be linear algebra, because without a grasp of vector representations of data and matrix operations on that data, a lot of statistical concepts, e.g. multivariate statistics, won't be very understandable. Fundamentally, trying to use statistics without this deeper understanding of the mathematics underlying much of it can lead to all kinds of issues, like choosing the wrong statistical method for a given problem.
- mycologos 3y ago
  I think a "statistics" class could still help a lot of people without ever touching continuous distributions (which as you mention are more naturally built off calculus) or multivariate distributions (which as you mention are more naturally built off linear algebra). Simple stuff like basic conditional probability, Bayes' rule, confidence intervals, basic hypothesis testing, and general "statistical thinking" don't really require calculus or linear algebra. Just getting people to a point where they understand some of the really basic tools seems pretty valuable, and pretty far from the current state of things.
  - lahvak 3y ago
    A mandatory nitpick: you cannot do conditional probability without multivariate distributions, but you are right, you do not need linear algebra for a basic introduction. Even continuous distribution can be introduced, you just start with discrete cases, introduce areas in a histogram, and then approximate the histogram by a continuous curve. You can even get to the Riemann integral that way if you want to. And nobody actually calculates normal probabilities by integrating the Gaussian function, it's kind of hard to do.
addicted 3y ago
Basically what we’ve found is that students need a better understanding of statistics, visualizations, etc. But just because these are mathematical concepts why do they have to replace math in the curriculum? Logically, shouldn’t they replace the least valuable parts of the curriculum, irrespective of whether that’s math or not? So the question really shouldn’t be statistics vs calculus (or advanced algebra) but rather statistics vs whatever is considered the weakest part of the curriculum across the board, not just limited to the field statistics belongs to.
- bambataa 3y ago
  Peak HN. Where do you find this shared view of what is considered the least valuable part of the curriculum?
ApolloFortyNine 3y ago
I took both ap statistics and ap calculus, got a 4 on statistics and a 5 in calculus, and in my opinion the calculus was way more helpful in college. But at the same time, I didn't feel like I was unprepared for college math. And I got a degree in an engineering field at Georgia Tech, so I don't think the classes were just easy or anything. So in my opinion people have to take a look at the standardized test scores, and not just 'this teacher chose to give them an A'.
- brightball 3y ago
  Yes. MIT came out and pushed back against all of the anti-standardized testing narrative to show the data. At least for math, those standard scores are incredibly reliable indicators of success.
  - otoburb 3y ago
    Interestingly, Caltech extended their standardized testing moratorium for another 3 years[1] out to 2025. Wishful thinking on my part, but I wonder if the two admissions departments would ever get together and compare notes, maybe even in a public forum. [1] https://www.admissions.caltech.edu/apply/first-year-applicants/standardized-tests https://www.admissions.caltech.edu/apply/first-year-applican...
    - brightball 3y ago
      Here’s the MIT info if you’re curious. https://mitadmissions.org/blogs/entry/we-are-reinstating-our-sat-act-requirement-for-future-admissions-cycles/ https://mitadmissions.org/blogs/entry/we-are-reinstating-our...
      - otoburb 3y ago
        Caltech at least will continue studying the performance of their cohorts through the moratorium extension (including a graduating class). I find it fascinating that two of the supposedly most rigorous academic American STEM undergraduate programs arrived at completely different conclusions from their internal studies (thus far).
dhosek 3y ago
I’ve always felt that the best capstone high school class is neither stats nor calc, but finite math, which ends up being a mix of some linear algebra, combinatorics and probability with a dash of discrete mathematics. This is stuff that has a lot more applicability in business and social science and is under-taught (one example was one that came up when I was getting my MS/teaching credential was my fiancée’s step-father wanted to be able to estimate the length of a roll of sandpaper given the diameter and thickness of the sandpaper. The engineers at the company came up with a correct answer by writing a program to add up the lengths of each layer of sandpaper, I verified their answer with a bit of pencil-and-paper algebra impressing my future in-laws. Someone who had taken a finite math class would¹ be able to do the same. ⸻ 1. Although perhaps I’m being optimistic since I’ve neither taken nor taught finite mathematics, I’ve only paged through a textbook I saw in a bookstore once.
atleastoptimal 3y ago
A lot of people talk about adult mathematical/statistical illiteracy and how it leads to major problems with cognitive biases, poor decision making, etc. However I haven't seen anything that explicitly ties having had a class in X leading to that person applying X more frequently in their life. The breakdown as I imagine it is: 1. The top 10-20% of students implement X vaguely into their life 2. The bottom 80-90% merely take the class to pass and forget about X and fail to apply it to their life. They are still prone to cognitive biases because they haven't extrapolated the effectiveness of what they've learned to the real world. I think generally, at least in America, a general culture of anti-intellectualism prevents people from actively caring about what they learn in school and applying it to the real world. It's not a mere matter of education, it's about making education meaningful to people over the alternative race to the bottom.
- lupusreal 3y ago
  Related, there's no evidence that people behave more ethically after taking a course in ethics. But the education system has one hammer, teaching courses about subjects, so every problem looks like a nail. They want their students to behave ethically so for school systems there can be only one solution; they teach ethics courses. 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.
  - hnfong 3y ago
    Reminds me of SBF's mum being a professor in legal ethics and moral philosophy...
addicted 3y ago
The worst part about these discussions is that no one actually remembers what they learn in high school unless they build upon it further in college (or it’s something they’re more likely to encounter it in day to day life, such as history, geography, literature, etc). The real purpose of education, in high school but in general until your junior year of college in the American system at least, is to learn how to learn. And advanced math involves the most unique and mind challenging ways of learning in a high school education. It involves memory, application, pattern recognition, etc but it’s about the only part of the high school curriculum that encourages abstract logical thinking. Literature does this too, but it doesn’t require it, and it doesn’t require the rigor in abstract thinking that’s needed in advanced math. In an argumentative essay you could hide flawed thinking with wordplay and any ways the logical accuracy of the argument only forms a small part of the grade but with advanced math there is no place to hide flawed logical thinking…there being a single correct answer, and the fact that math is a language that is designed first and foremost to eliminate ambiguity, means it’s hard to get away with sloppy thinking. Advanced math that isn’t necessarily useful in your college career is absolutely essential to teach high quality thinking. A statistics course, on the other hand, will almost certainly devolve into the practical and the useful, so much like high school physics will not teach thinking but simply knowing how to apply the right formula at the right time. That’s a useful skill but one that’s taught all over the current curriculum. 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.
- jasperry 3y ago
  > Advanced math that isn’t necessarily useful in your college career is absolutely essential to teach high quality thinking. 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.
gnicholas 3y ago
> But Martin said that non-STEM students didn’t really need to learn trigonometric functions, which are used in satellite navigation or mechanical engineering. For me, trig has been the least useful math. I never once used it in college (2 years as a CS major, then an Econ major) or in the working world (tax lawyer, startup founder). I'm sure there are some jobs where it is irreplaceable, but this might be a minority of STEM jobs. Do biologists need to know trig?
- ykonstant 3y ago
  I once TA'd a Calculus for Business course; there were two fundamental differences between that and the regular course: 1) the problems and exam material were much easier and 2) it lacked any mention of trigonometric functions. 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.
  - hnfong 3y ago
    Ironically they can only do that without the trig functions. Imagine asking somebody on Bloomberg how they would model and quantify the "economic cycle" using trig. :D
    - ykonstant 3y ago
      "W..what do you mean exponential growth is not the only dynamic?!"
      - tgv 3y ago
        Nah, they slap a few linear variables on it, throw that in a model that adds lag/time dependent variables, and they're ready to tell you the pendulum is starting to swing in the other direction.
lessbergstein 3y ago
One of my freshman classes was game theory. A prerequisite was trigonometry, and I only had pre-Calc. The best part about it was I was doing averages and which great which number is greater in my senior high school class after taking pre-Calc as a Sophomore. Of course I complained about this to both schools but that wasn't really their problem. I suppose they just didn't care.
diffeomorphism 3y ago
> Majors that require calculus were excluded. So they asked exactly those majors that need less calculus and more statistics whether high schools should teach less calculus and more statistics. > College professors were more keen on an assortment of what was described as mathematical “practices,” The fact that this is phrased as unexpected is quite sad: https://en.wikipedia.org/wiki/A_Mathematician%27s_Lament https://en.wikipedia.org/wiki/A_Mathematician%27s_Lament School is supposed to be about learning, reasoning and developing skills and not just about memorizing or checking a list of topics.
baazaa 3y ago
This always degenerates into a calc vs stats debate when really the problem is the vast majority of the calc curriculum is a waste of time. Even if you become a professional engineer, you're not going to be solving lots of complicated integrals by hand. If the curriculum started with 'imagine the student has access to a computer' you could easily cover not just stats and calc but a whole lot of discrete math etc. as well, just from the time saved not teaching everyone the huge 'bag of tricks' necessary to solve a few equations by hand (in the real-world most things don't have closed-form solutions so the tricks are utterly pointless). Note the problem goes far beyond high-school curricula. I studied a subject on queuing theory at one point and realised that everyone in the class was less equipped to deal with any real-world queuing problem than any of my CS friends who could code. Simulating the queue allows you to solve most things, whereas only an extremely narrow class of problems could be solved by hand. The only reason the maths curriculum is the way it is is because it's barely changed since computers were invented.
- hnfong 3y ago
  The teachers who grew up without computers wouldn't change the curriculum because they wouldn't see the point. 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. --- 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.
martopix 3y ago
I teach in a high school that is very humanities-focused (and the students explicitly want that, they chose it). I realized statistics was not taught, and I introduced a module on it for my classes. I thought it was insane to do three chapters on trigonometry, including solving useless equations with goniometric functions, and doing zero stats and probability. These students will likely (and statistically) choose uni careers such as medicine, psychology, law, politics -- the ones who choose engineering are very rare. Statistics is THE ONE field of math that they need.
ojosilva 3y ago
I would only teach applied math in high school, not the Applied Math curricula from most universities, but a vertical that covers things that can be made sense without depth, just breadth. I'd leave most abstractions and numeric solving for later in life or STEM students. I just feel I did so much equation solving for nothing in my student life! I remember too many times the teacher would assign the equation solving exercises 1 - 10 as homework, and only problems 11 and 12, leaving problems 13 to 20 out. Why? Well, problems are hard to teach, to grasp and to solve! Most math is actually very easy if kept easy, even integrals only have a few rules. Don't make a student do 700 variations of integral solving. Find a practical use for it (ie find the satellite velocity and position), introduce the subject with the problem "Today we're going to find a velocity of a satellite, there's a thing called integral that helps us do that, this is why and how..." and do that problem 20 times over and over. A variety of real-world problems have very simple math to solve, like stress and strain in materials engineering, center of gravity in physics, etc. Problem-solving is the worst of all math for most kids - so do repetitive problems and teach kids to extract information from problems into formulas and equations. And, yes, do plenty of statistics which is rich in real-world use too. Having a problem-solving, math applying mind and probabilistic view of the world are the skills to have at any field!
bluenose69 3y ago
One line of the article stood out for me: "College admissions officers value calculus, almost as a proxy for intelligence." I think what lack of Calculus signals is more lack of facility with Mathematics generally. When I speak with students who have weak (or missing) Calculus backgrounds, I usually find that they are uncomfortable with something much more basic: they simply do not understand what it means to use a symbol for a quantity. Not understanding that, they don't see how manipulating symbols (e.g. subtracting something from both sides of an equation to move a term to the other side) makes any sense. It's as though they had a mental block on the day when a teacher said "let x be the unknown". They usually bluffed their way through that class, and the next and the next, as they proceeded through middle school. But they never really got their heads around the ideas. And this, not intelligence, is the problem with their later success in STEM fields. Helping such students is a real challenge. It's a matter of establishing a connection and a trust that will let you probe back into their past until you find the place where the problem arose. This is like psychotherapy. It takes one-to-one work and it takes a long time to build trust, before the probing can begin. None of this is practical in a traditional college teaching framework, and that is why college admissions offers key on Calculus. As for the discussion of Statistics, I agree that this is more important for general students. STEM students need to add Calculus as well.
- zozbot234 3y ago
  > (e.g. subtracting something from both sides of an equation to move a term to the other side) 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.
OJFord 3y ago
How much time roughly do you spend in math classes in school in the US? I'm finding it a bit baffling reading these comments ('calculus vs statistics' etc.) from my UK perspective. We had I think roughly^ 3x1h maths lessons each week, and it didn't seem to be a problem to do both, as well as whatever else. Nor did we wait until secondary school to start stats (calculus sure, for the usual colloquial definition of it). Or are we talking specifically about some arbitrarily advanced level of each that somehow everyone's on the same page on without describing? (^it's been a while!)
wkjagt 3y ago
I’ve read a couple of texts by Paul Lockhart. His opinion (which I support) is that what is taught in high school as math isn’t math, as in what a mathematician does. As an example: Pythagoras proved the relation between the lengths of the sides of a right triangle. The journey towards finding the answer to this is math. Or even exploring why this relation even exists based on his findings is math. The answer to the question is expressed as a2+b2=c2. What we’re asking students to do is just to take that formula and apply it hundreds of times without going into the actual math behind it. This, he says, isn’t math at all. The whole “math” step is skipped. He compares it to if for example we were teaching music in school without ever touching a musical instrument, and just learning how to read and write notes. Or teaching art without ever touching a pencil or brush. The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. The same way we could totally miss out having students never discovering their artistic talents if art class was taught without ever actually doing any art. His text that explains this most clearly is “A Mathematician’s Lament”. It’s a great read and not very long.
- j7ake 3y ago
  Society thinks of math as technical and functional, rather than beautiful. 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.
  - TeMPOraL 3y ago
    I disagree somewhat with both you and GP here: > Society thinks of math as technical and functional, rather than beautiful. The math people suck at is the technical and functional part, just as much as the beautiful parts. I'd frame the issue like this: there's the journey of arriving at the result, there's wielding the result, and then there's applying it to a very particular problem. Schools are teaching mostly the last part - applying a^2+b^2=c^2 to a set of contrived problems that are meant to exercise your ability in arithmetic, variable substitution, and simple symbolic transformation. They do not teach people how to wield the formula - as in, how can you realize on your own that any particular real-life challenge calls for application of that tool (or of mathematics in general). That's the middle part that's directly valuable, in the immediate term. That's the difference between a carpenter, and a person who figured out how to use a hammer to drive a nail into a wall. The beauty and journey part? That's study of history and art of inventing new types of hammers and other impactors, and perhaps inventing your own. Even further upstream of the "knowing how to use it to build something" part, but neither is taught in schools.
MarkusWandel 3y ago
Old fart here: In high school, math was broad spectrum. Calculus, Algebra, Geometry etc. Too early for specialized topics like Probability and Statistics. Those were first-year university material, where you start getting steered toward the actual math skills you'll need for your curriculum. What's wrong with that?
nameless912 3y ago
I felt this article deep in my old creaky bones. I hated math in middle school and high school. This is despite loving science, and enjoying math all the way through Algebra 1. However, when I started taking Algebra 2 and beyond, it seemed like we shifted away from why things work to solving ever larger polynomials and other tedious tasks, getting points taken off along the way for a flipped sign or digit or moving a decimal point, even when the entire rest of the calculation was correct. I ended up doing a degree in computer engineering, and every class that involved algorithm design or programming I excelled at, while every "hard math class" (for example, statistics, electromagnetism, etc.) I barely passed. I didn't fall back in love with math until I took discrete mathematics and signal processing in the same semester, and then later a class on algorithms and formal models of computation. They showed me that math was actually beautiful and fascinating, an art form unto itself. These three classes showed me how amazing math can be; I remember particularly that the day we learned about FFT's my brain felt like it had ascended into another dimension. They truly felt magical. Same thing with learning about recurrence relations and finally "getting" dynamic programming. Now I love math, and I am planning on taking this holiday break to finally crack open the Principia Mathematica and trying to really understand it (a personal goal of mine for several years). I can't wait to share the joy of mathematics with my kids when they're old enough. I just hope that they don't get bogged down in the mechanical, number crunching part like I did and manage to continue to see the beauty of the underlying ideas.
PM_me_your_math 3y ago
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bikenaga 3y ago
Some comments about points people have raised ... 1. Can you teach statistics without doing calculus first? It's quite common. We had such a course which was typically taken by social science majors who needed some statistics in their fields. (I never taught it.) The prereq was just a little algebra. A typical text was Mario Triola's "Elementary Statistics" (see here for the table of contents: https://www.amazon.com/Elementary-Statistics-13th-Mario-Triola/dp/0134462459 https://www.amazon.com/Elementary-Statistics-13th-Mario-Trio...). The students used Minitab as part of the course. But we also had prob & stat courses which used calculus. 1. (a) How can you define the exponential function (or the number e) without calculus (limits)? When I taught college algebra (= review of high school algebra, around two courses below Calc 1) I'd say something like: "There's a number called e which is approximately 2.7118281828459045 ... - it's kind of like pi, an infinite decimal that never repeats. We've seen there are exponential functions like 2^x, and we have an important exponential function called e^x. You're probably wondering why we're using such a weird number in an exponential function. You'll see how it comes up if you go on to take calculus." No one much complained about any of that. Even a typical calc course skips a lot of the theoretical background - you don't see the background unless you take a course in real analysis. In teaching college math, you're always "starting in the middle", assuming a lot, and in basic courses you're omitting a great deal of the rigor. (Also, in calculus it's common to define logs first using a definite integral, then discuss inverse functions, then define e^x as the inverse of ln x [so in particular e is just e^1]. You justify the notation by showing ln x and e^x have the properties you "expect" logs and exponentials to have [e.g. ln (a b) = ln a + ln b].) 2. Why not teach statistics/linear algebra/logic in high school? High school curricula are often strongly determined by state testing, and by the needs of students who are going on to college to have the math that colleges expect. I took courses in math logic and matrix theory in high school, but they were electives; the "big" course for me from the point of view of college admissions was BC calc. I certainly got no college credit for the logic or matrix theory courses. So e.g. if you tried to do linear algebra or logic in high school it would come at the expense of topics that a state is testing - then your students do poorly (even though they might know some pretty good math) and you and your school get in trouble. There is more of a movement to teach stat and data analysis in high school and college - things are changing. 3. Why not teach statistics/linear algebra/logic before/in place of calculus in college? College math departments are heavily service departments. The bulk of the credit hours are in courses taken by majors from other departments. Those departments (science, engineering, business, and so on) tell math departments what their majors need, and math department have to ensure that the math courses suit. The other departments in turn are often required to have majors take certain courses with certain topics by program accrediting agencies. So there can be a lot of inertia there.

143 points by bikenaga 3y ago

I have a kid starting 1st grade and I have just been exposed to common core math for the first time. At a glance I do not understand how it prepares for a future in mathematics or related fields. Interested in hearing from others who have been through common core math first hand or raised a child through the program Also the way i learned to divide in my head is completely backwards and, though it works for me, when i tried to explain it to my wife i realized how completely backwards and inefficient it is, so maybe common core helps with these and other hard-to-internalize concepts?

rahimnathwani 3y ago
When you say 'common core math' are you referring to the standards themselves, or a particular set of textbooks that cover the required material?
koolba 3y ago
> Also the way i learned to divide in my head is completely backwards and, though it works for me, when i tried to explain it to my wife i realized how completely backwards and inefficient it is, so maybe common core helps with these and other hard-to-internalize concepts? What way? The classic way to learn is to be proficient at multiplication up to 10x10 and then solve for the reverse.
- deleted 3y ago
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irrational 3y ago
Well, one thing is that I can’t help my kids with their homework. I simply don’t understand the way they do the math these days. They get the same answers, but go through entirely different steps. I remember one parent teacher conference where my kid was in attendance (4th grade). The teacher was showing my kid how to do a problem. I was able to get the answer in my head, but she was using all these boxes and things to get the answer and I could not follow along. I still have no idea how she got the right answer doing that. It seemed incredibly complicated.
- practicemaths 3y ago
  I do not have a child or direct experience with common core. Personally I suspect issues lie more in teachers themselves not being taught correctly than the actual concept behind common core though. From what i can gather it is trying to introduce the concept of factoring earlier which in my opinion makes sense since that is really useful in Algebra and above.
  - j45 3y ago
    Learning to teach is very different than learning, let alone first principles learning for children. Having basic understanding, then skills, then competency to remember and test via assessment in that order can be missing even for graduates. Universities focus on teaching young adults how to teach children. The rest of the world as adults don’t learn how children do.
    - practicemaths 3y ago
      And many teachers have continual education. That means the implementation of common core is also upon teachers with decade(s) of experience being told to teach differently. Regardless, if your child has a nack for maths that is a great advantage over the other children and worth investing in since so few people bother to learn it at all. Additionally it leads to better reasoning and logic in other academic and professional settings as well. Which means a better chance at a higher income and ability to sustain and secure their livelihood and thus afford having their own children possibly. Which means a better chance for continuation of a little bit of yourself.
- monocasa 3y ago
  The point of most of the common core math changes were to teach numeracy rather than rote arithmetic. Remember how some teachers said "you won't just have a calculator on you at all times" and it turned out that we all carry around what would have been supercomputers for the time? Early math education realized that too, and shifted the focus to quickly estimating solutions through understanding the relationships of their numbers so you can go "hey, something's not here" when you work with your calculator, rather than just accepting whatever garbage out came from your garbage in. It is complicated, but the point isn't to get the right answer per se, it's to give kids a geometric intuition of numerical relationships by going through the whole rigamarole.
analog31 3y ago
My kids are now college age, but when they were in K-12, I looked up common core, since the standards are published in several states. Honestly it looked quite a lot like the math that I learned in school, give or take. I took exactly the same school math subjects. 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.
- calvinmorrison 3y ago
  for the general populace, good arithmetic, with a smattering of practical geometry and trigonometry should be the focus for more years of education.
secabeen 3y ago
Primary school common core math is focused on teaching numeracy and conceptual understanding. Students who want a future in mathematics or related fields will take more formal rigorous math in secondary school, often via a "compaction" program that resembles older math programs that result in completion of Calculus by the 11th or 12th grade. 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.)
- zozbot234 3y ago
  Repetitive practice is important - it's about getting exposure to formal reasoning in a "toy" context. You can't get "conceptual understanding" in math without learning what formality and rigor is for.
currymj 3y ago
i think you're exactly right -- it is trying to take some of the mental tricks that "good at math" students would intuit for themselves, and teach them explicitly. this is good overall, although of course it's possible to ruin anything via bad teaching.

Scubabear68 3y ago

I want to know what world this author is from. High School math in public schools has been dumbed down repeatedly over the years. Calculus in High School? Increasingly rare to non-existent. What we need is the opposite - a return to traditional curriculums, and stop non-sense like a 5 credit course in graphic novels (a thing in our district school).

threeseed 3y ago
Why is that course nonsense if it's elective ? Calculus is irrelevant for the majority of students.
- Scubabear68 3y ago
  It’s nonsense because this is a public high school, government mandated and paid for by taxes.
  - ceejayoz 3y ago
    Helping kids engage with literature they enjoy seems a lot worthier of my tax dollars than, say, teaching cursive.
    - BlackjackCF 3y ago
      Please tell me they don’t teach cursive anymore. I remember being fed the lie that I would have to use cursive as an adult. I don’t even really write anymore…
      - ceejayoz 3y ago
        They do still teach cursive, at least here in New York state.
      - ern 3y ago
        Cursive is making a comeback. Google it. The education establishment is often divorced from reality.

hn_throwaway_99 3y ago

This is essentially the "calculus vs. statistics" debate that has been going on for some time now. I'm definitely on the side of making statistics mandatory, and then making pre-calc/calc electives for motivated students who want to enter STEM fields. Primary reason for this is that I believe a base knowledge of statistics is so important for everyone in modern society, and it directly goes to better being able to evaluate the constant flood of information we are now all subjected to.

deleted 3y ago
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fasthands9 3y ago
I agree - as long as it's not an excuse to dumb things down statistics should be taught before Calculus. It's a quirk of history, I think, that we have it the way it is. Statistics is still going to be important to understand for 99% of STEM majors, so I don't think there's much downside to flipping the order.
- Doozy7685 3y ago
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cowsandmilk 3y ago
AP Statistics has existed since 1996 and my high school (and many others) offered it as a path for students in addition to the calculus path. That is, you could head down the statistics path or the calculus path (or both) depending on your interests.
- smeej 3y ago
  I finished the AP Calc path my junior year, but was told colleges would still expect to see a 4th year, so I signed up for AP Stat. It was mind-numbing how easy it was by comparison. We got to the end of the semester and I realized we'd only just reached some of the principles from the first week of calculus. I don't remember why he was so proud to point out that we could now calculate derivatives (it was a long time ago), but I was thoroughly unimpressed. By the second semester, I had my college acceptance letter, so I didn't bother with the other half.

kurthr 3y ago

Maybe history, social studies, government, or economics classes in highschool should use a spreadsheet to show basic data analysis and statistics (mean, median, mode, stdev). It would be good if there was a more basic understanding of how an "average" can hide a lot of inequality. I don't really see getting rid of angles and algebra to cover something "outside of STEM".

nemo44x 3y ago
You assume the people teaching those subjects could do that.
starcraft2wol 3y ago
> (mean, median, mode, stdev) 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.
- dagw 3y ago
  This is where taking a more holistic approach to curriculum planning would pay off. If the weeks after you were taught these (or any other) concepts in math class they start showing up in your science and geography classes, you'd probably remember them better and have a better understanding of how to use them.

nebula8804 3y ago

Man oh man does this bring back past trauma. I graduated high school in 2006 and despite me being in one of the top half rated high schools in my state and having a consistent B average, I was not prepared at all for when I entered my engineering school (NJIT). A lot of my math was lacking fundamentals and instead was just memorization and 'plug and chug' style learning. That does not fly at university especially an engineering school. Schools (at least my school) really expected you to learn fundamentals as their tests would not just be a variation of the problems taught in the lectures but a unique twist on the concept itself. The school tested me on my skills before entering and put me in a 'remedial' course called Pre-Calculus with a Calculus Perspective. They even wrote a custom textbook to really fine tune getting prepped for what was to come. (Calc 1, 2, 3, Diff-Eq). I wasn't enough because the new thought process requires a great deal of effort to internalize in the timeframe given. I was barely treading water my first semester and ended up failing math and physics and got put on probation. I eventually graduated and am doing quote ok in life career wise but it was an extremely painful ordeal. Not being prepared and really ready to go remains to this day one of the biggest regrets of my life. I really feel like I let down the people who made my education possible (parents, teachers, schools). Sometimes I wonder why is there such a dumbing down of mathematics in the US? The US is powerful and at the top of their game because of science and technology. Why try your best to kill your golden goose?

jaybrendansmith 3y ago
Similar story here. What I learned in math in high school was an absolute joke. It in no way prepared me for physics and math in college. In addition, most college majors do not need it, they need an understanding of statistics, and a much better understanding of 'business math' and algebra. I think dividing kids into two or more tracks does not hurt them, because they can always take extra course work in college if they ultimately decide they want to pursue STEM. Where it hurts them are the standardized tests, this seems to be the issue.
tayo42 3y ago
> instead was just memorization and 'plug and chug' style learning. 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.

smeej 3y ago

I have a friend who has been a high school math teacher for 15 years. When she started, she asked me and the rest of our friends to make note of any times we use algebra or more advanced high school math in our daily lives. Both times it has happened, I've been sure to let her know. Once, the Pythagorean theorem helped me figure out how long the sides of square table decorations needed to be for my sister's wedding if they were to be inset in round tables with 5-foot diameter. The other time, I was able to prove, conclusively, that I could not possibly have been speeding before another driver caused an accident, but none of the police officers could follow the math, and since I wasn't a "certified accident reconstruction expert," they wouldn't believe me anyway. (Hiring one would have been more than the repair cost.) It didn't go over well when I told the supervising officer that just because he couldn't do the math, it didn't mean it couldn't be done, even by some dork who had a liberal arts degree but had, in fact, studied calculus in high school.

j45 3y ago
One thing that stood out in university was learning the algebra version to so many basic math formulas in any field, whether it might be physics, math, biology, chemistry, or most anything. As for your use case, sometimes, some people are hired to not understand things.
hutzlibu 3y ago
"The other time, I was able to prove, conclusively, that I could not possibly have been speeding before another driver caused an accident" Can you share that proof?
- smeej 3y ago
  I don't still have it. It was years ago. I had the accident record pulled from the vehicular equivalent of my car's black box that said how fast I was going at impact. The distances involved made it such that I could not have braked to that speed from a higher speed than the speed limit in the required space.

j45 3y ago

College doesn’t often enough prepare a graduate for a career in their field of study. Math is important because it underpins so many decisions improving a tiny bit better over a long period of life, if you can measure and compare. Without it is often more profitable for those who want profit. edit: clarity

nemo44x 3y ago

How about we start with high school students being prepared for math that concerns their adult life. Like how debt works, compound interest, savings, budgeting, etc?

smeej 3y ago
Reminds me of that joke that crops up in April, about how grateful we are for all those hours in our high school math classes this parallelogram season.
- nemo44x 3y ago
  Geometry and abstract math is important. But getting the practical stuff into even remedial heads is necessary to. Should do both.
jmye 3y ago
We had home ec in junior high and high school - students just chose not to take it. I’d add that those are 3rd grade “math” concepts - not replacements for actual math courses.
SamuelAdams 3y ago
Many schools do teach this. It’s not required but some schools do offer it as an elective. However, as practical as it may be, many teenagers just aren’t interested in learning about personal finance. 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.

jccalhoun 3y ago

I teach Communication at college but I was a math minor in undergrad. I hear my students complain about math class all the time, how they hate math, and it isn't rare to hear students say they have taken the same math class multiple times. I've glanced at the math book and it is basic stuff. I don't know how but we need to get kids to not hate math before we can do anything else.

tayo42 3y ago
I think its because of how its taught. one thing is its often for a lot of people just learning for the sake of learning once your doing anything like algebra or harder.
deleted 3y ago
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kwk1 3y ago
"A Mathematician's Lament" by Paul Lockhart is a pretty good read (25 pages) on the topic, I think: https://archive.org/details/AMathematiciansLament https://archive.org/details/AMathematiciansLament

hutzlibu 3y ago

Data point, (or rant) about high school (Abitur) situation in germany: it is the same or worse. I was quite good in math in school, yet in the beginning did not understand anything math related in university. Which was no surprise, because the math in school for me was basically memorizing algorithms to solve known problems of style X. But not really understanding it. And then in the final years in school, while in theory doing higher math, the main focus shifted on how to use a graphical calculator to do the math for us. But in university I suddenly had to understand what I was doing, to be able to use it. Oh my. So it was just really hard grind, to recover all that I did not learn before (that grind was highly beneficial, though). In theory the universities knew about the problem and they complained about it to us - but as far as I know, nothing has changed since then. So Math will remain a weeding factor for lot's of students, who otherwise might have been great engineers, but maybe had the bad luck, of having bad math teachers on top of the bad curriculum. It is also a really bad sign, that math is so universally hated among students. Because math done right, is just thinking done right - and I think the world might benefit from more of it.

squirrel6 3y ago

Honestly, the bigger issue here is that good math teachers in the US are just in short supply. Sometimes I wonder if the way we teach algebra here is flawed, like something pedagogically or conceptually, and good teachers break through that to give students what they need to prepare their students.

bigger_cheese 3y ago

Even in the STEM world I don't think statistics is taught all that well. I studied Engineering a STEM subject, this was sometime ago (2000's) and at that time we were required to take a "statistics for engineers" course in the second year of the degree. The subject was mostly useless (it pretty much boiled down to "plug this data into Excel and this is how you interpret the ANOVA output"). What I really wish they focused on was regression analysis and especially non-linear regression, logistic regression etc. I had to learn all about it in the workplace after graduation. Even the statistics knowledge we were taught was incomplete - for example I did not learn about Cross-validation at university. talking to more recent graduates I don't think the situation has improved all that much in the intervening years.

defrost 3y ago
The "STEM World" is a large one and "XXX for Engineers" courses in undergraduate university are long on material to rote memorise and short on depth required to understand and expand. 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.

nobodyandproud 3y ago

How exactly does pre-calc really prepare a student for calculus? I remember taking Trigonometry—useful—-and then taking pre-calc.

akasakahakada 3y ago

Funny that most humanities degree holders that I am aware of do not agree that humanties events can be expressed in numbers. They can't comprehend anything rather than over simplified stories. What I mean by degree holders, I mean Phd degree holders, professors. What being said in the article reflects misunderstandings of math from the arts and humanities community. What they need is a general understanding of discrete math rather than specifically statistic. Calculus is valuable to STEM because of the smooth and continuous natural of macroscopic world. Contrast that cultural/social/econ events are discrete. If you want to draw a beautiful smooth line which fit so badly, that do not tell the whole story. However we see people drawing line cut through a circle in a graph in most humanities papers and they claim there is a positive relationship whatsoever. And here is how flawed the survey is. >>> Survey target: Majors that require calculus were excluded. >>> Many high school math topics were unimportant to college professors. For example, most professors said they wanted students to understand functions, particularly linear and exponential Linear functions are already taught in pre-calculus. You want calculus but you don't want calculus. >>> the ability for patterns and relationships and make generalizations This is unrelated to math. More related to IQ. Seems that students in the non STEM realm generally lack the ability to find patterns as professors witnessed. People should take more concerns on this phenomenon.

ajsnigrutin 3y ago
> This is unrelated to math. More related to IQ. Seems that students in the non STEM realm generally lack the ability to find patterns as professors witnessed. People should take more concerns on this phenomenon. 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.

dboreham 3y ago

US-centered response, assuming the article is about US education. The article is off track vs what I observed with my kids math (typical middle of the country small public hs). Nobody gets to do trig until they've done boat loads of stats. The thing I saw missing was the kind of top notch teacher that can inspire and intrigue the students. This can only be addressed by paying teachers much more. That's not going to happen. Our school only had one "real" math teacher, and he retired last year.

presentation 3y ago
Definitely ought to be paid more but I suspect even with more pay, most teachers don’t have that special sauce.
ApolloFortyNine 3y ago
Paying more goes hand and hand with testing teacher (or holding them responsible for students grades), and generally that's not very well received by teacher unions. 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.

photochemsyn 3y ago

I don't understand the statistics-vs-calculus debate at all. Probability theory is at the heart of statistics, and how can a student understand statements like 'the integral of a probability density function must equal one over its possible range' without taking some calculus? The problem with calculus as taught today in most schools is that most of the mathematical tricks for integrating functions that they hammer the students with are essentially useless in real-world problems so you might as well just start with numerical methods of approximation right at the beginning, once basic concepts like 'the integral of a function corresponds to the area under the curve of that function' are grasped. I know many math purists are dedicated to the pencil and paper approach, but it's not all that useful for non-math majors who will certainly be building or using computational models almost exclusively. Another prerequisite for statistics should be linear algebra, because without a grasp of vector representations of data and matrix operations on that data, a lot of statistical concepts, e.g. multivariate statistics, won't be very understandable. Fundamentally, trying to use statistics without this deeper understanding of the mathematics underlying much of it can lead to all kinds of issues, like choosing the wrong statistical method for a given problem.

mycologos 3y ago
I think a "statistics" class could still help a lot of people without ever touching continuous distributions (which as you mention are more naturally built off calculus) or multivariate distributions (which as you mention are more naturally built off linear algebra). Simple stuff like basic conditional probability, Bayes' rule, confidence intervals, basic hypothesis testing, and general "statistical thinking" don't really require calculus or linear algebra. Just getting people to a point where they understand some of the really basic tools seems pretty valuable, and pretty far from the current state of things.
- lahvak 3y ago
  A mandatory nitpick: you cannot do conditional probability without multivariate distributions, but you are right, you do not need linear algebra for a basic introduction. Even continuous distribution can be introduced, you just start with discrete cases, introduce areas in a histogram, and then approximate the histogram by a continuous curve. You can even get to the Riemann integral that way if you want to. And nobody actually calculates normal probabilities by integrating the Gaussian function, it's kind of hard to do.

addicted 3y ago

Basically what we’ve found is that students need a better understanding of statistics, visualizations, etc. But just because these are mathematical concepts why do they have to replace math in the curriculum? Logically, shouldn’t they replace the least valuable parts of the curriculum, irrespective of whether that’s math or not? So the question really shouldn’t be statistics vs calculus (or advanced algebra) but rather statistics vs whatever is considered the weakest part of the curriculum across the board, not just limited to the field statistics belongs to.

bambataa 3y ago
Peak HN. Where do you find this shared view of what is considered the least valuable part of the curriculum?

ApolloFortyNine 3y ago

I took both ap statistics and ap calculus, got a 4 on statistics and a 5 in calculus, and in my opinion the calculus was way more helpful in college. But at the same time, I didn't feel like I was unprepared for college math. And I got a degree in an engineering field at Georgia Tech, so I don't think the classes were just easy or anything. So in my opinion people have to take a look at the standardized test scores, and not just 'this teacher chose to give them an A'.

brightball 3y ago
Yes. MIT came out and pushed back against all of the anti-standardized testing narrative to show the data. At least for math, those standard scores are incredibly reliable indicators of success.
- otoburb 3y ago
  Interestingly, Caltech extended their standardized testing moratorium for another 3 years[1] out to 2025. Wishful thinking on my part, but I wonder if the two admissions departments would ever get together and compare notes, maybe even in a public forum. [1] https://www.admissions.caltech.edu/apply/first-year-applicants/standardized-tests https://www.admissions.caltech.edu/apply/first-year-applican...
  - brightball 3y ago
    Here’s the MIT info if you’re curious. https://mitadmissions.org/blogs/entry/we-are-reinstating-our-sat-act-requirement-for-future-admissions-cycles/ https://mitadmissions.org/blogs/entry/we-are-reinstating-our...
    - otoburb 3y ago
      Caltech at least will continue studying the performance of their cohorts through the moratorium extension (including a graduating class). I find it fascinating that two of the supposedly most rigorous academic American STEM undergraduate programs arrived at completely different conclusions from their internal studies (thus far).

dhosek 3y ago

I’ve always felt that the best capstone high school class is neither stats nor calc, but finite math, which ends up being a mix of some linear algebra, combinatorics and probability with a dash of discrete mathematics. This is stuff that has a lot more applicability in business and social science and is under-taught (one example was one that came up when I was getting my MS/teaching credential was my fiancée’s step-father wanted to be able to estimate the length of a roll of sandpaper given the diameter and thickness of the sandpaper. The engineers at the company came up with a correct answer by writing a program to add up the lengths of each layer of sandpaper, I verified their answer with a bit of pencil-and-paper algebra impressing my future in-laws. Someone who had taken a finite math class would¹ be able to do the same. ⸻ 1. Although perhaps I’m being optimistic since I’ve neither taken nor taught finite mathematics, I’ve only paged through a textbook I saw in a bookstore once.

atleastoptimal 3y ago

A lot of people talk about adult mathematical/statistical illiteracy and how it leads to major problems with cognitive biases, poor decision making, etc. However I haven't seen anything that explicitly ties having had a class in X leading to that person applying X more frequently in their life. The breakdown as I imagine it is: 1. The top 10-20% of students implement X vaguely into their life 2. The bottom 80-90% merely take the class to pass and forget about X and fail to apply it to their life. They are still prone to cognitive biases because they haven't extrapolated the effectiveness of what they've learned to the real world. I think generally, at least in America, a general culture of anti-intellectualism prevents people from actively caring about what they learn in school and applying it to the real world. It's not a mere matter of education, it's about making education meaningful to people over the alternative race to the bottom.

lupusreal 3y ago
Related, there's no evidence that people behave more ethically after taking a course in ethics. But the education system has one hammer, teaching courses about subjects, so every problem looks like a nail. They want their students to behave ethically so for school systems there can be only one solution; they teach ethics courses. 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.
- hnfong 3y ago
  Reminds me of SBF's mum being a professor in legal ethics and moral philosophy...

addicted 3y ago

The worst part about these discussions is that no one actually remembers what they learn in high school unless they build upon it further in college (or it’s something they’re more likely to encounter it in day to day life, such as history, geography, literature, etc). The real purpose of education, in high school but in general until your junior year of college in the American system at least, is to learn how to learn. And advanced math involves the most unique and mind challenging ways of learning in a high school education. It involves memory, application, pattern recognition, etc but it’s about the only part of the high school curriculum that encourages abstract logical thinking. Literature does this too, but it doesn’t require it, and it doesn’t require the rigor in abstract thinking that’s needed in advanced math. In an argumentative essay you could hide flawed thinking with wordplay and any ways the logical accuracy of the argument only forms a small part of the grade but with advanced math there is no place to hide flawed logical thinking…there being a single correct answer, and the fact that math is a language that is designed first and foremost to eliminate ambiguity, means it’s hard to get away with sloppy thinking. Advanced math that isn’t necessarily useful in your college career is absolutely essential to teach high quality thinking. A statistics course, on the other hand, will almost certainly devolve into the practical and the useful, so much like high school physics will not teach thinking but simply knowing how to apply the right formula at the right time. That’s a useful skill but one that’s taught all over the current curriculum. 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.

jasperry 3y ago
> Advanced math that isn’t necessarily useful in your college career is absolutely essential to teach high quality thinking. 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.

gnicholas 3y ago

> But Martin said that non-STEM students didn’t really need to learn trigonometric functions, which are used in satellite navigation or mechanical engineering. For me, trig has been the least useful math. I never once used it in college (2 years as a CS major, then an Econ major) or in the working world (tax lawyer, startup founder). I'm sure there are some jobs where it is irreplaceable, but this might be a minority of STEM jobs. Do biologists need to know trig?

ykonstant 3y ago
I once TA'd a Calculus for Business course; there were two fundamental differences between that and the regular course: 1) the problems and exam material were much easier and 2) it lacked any mention of trigonometric functions. 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.
- hnfong 3y ago
  Ironically they can only do that without the trig functions. Imagine asking somebody on Bloomberg how they would model and quantify the "economic cycle" using trig. :D
  - ykonstant 3y ago
    "W..what do you mean exponential growth is not the only dynamic?!"
    - tgv 3y ago
      Nah, they slap a few linear variables on it, throw that in a model that adds lag/time dependent variables, and they're ready to tell you the pendulum is starting to swing in the other direction.

lessbergstein 3y ago

One of my freshman classes was game theory. A prerequisite was trigonometry, and I only had pre-Calc. The best part about it was I was doing averages and which great which number is greater in my senior high school class after taking pre-Calc as a Sophomore. Of course I complained about this to both schools but that wasn't really their problem. I suppose they just didn't care.

diffeomorphism 3y ago

> Majors that require calculus were excluded. So they asked exactly those majors that need less calculus and more statistics whether high schools should teach less calculus and more statistics. > College professors were more keen on an assortment of what was described as mathematical “practices,” The fact that this is phrased as unexpected is quite sad: https://en.wikipedia.org/wiki/A_Mathematician%27s_Lament https://en.wikipedia.org/wiki/A_Mathematician%27s_Lament School is supposed to be about learning, reasoning and developing skills and not just about memorizing or checking a list of topics.

baazaa 3y ago

This always degenerates into a calc vs stats debate when really the problem is the vast majority of the calc curriculum is a waste of time. Even if you become a professional engineer, you're not going to be solving lots of complicated integrals by hand. If the curriculum started with 'imagine the student has access to a computer' you could easily cover not just stats and calc but a whole lot of discrete math etc. as well, just from the time saved not teaching everyone the huge 'bag of tricks' necessary to solve a few equations by hand (in the real-world most things don't have closed-form solutions so the tricks are utterly pointless). Note the problem goes far beyond high-school curricula. I studied a subject on queuing theory at one point and realised that everyone in the class was less equipped to deal with any real-world queuing problem than any of my CS friends who could code. Simulating the queue allows you to solve most things, whereas only an extremely narrow class of problems could be solved by hand. The only reason the maths curriculum is the way it is is because it's barely changed since computers were invented.

hnfong 3y ago
The teachers who grew up without computers wouldn't change the curriculum because they wouldn't see the point. 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. --- 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.

martopix 3y ago

I teach in a high school that is very humanities-focused (and the students explicitly want that, they chose it). I realized statistics was not taught, and I introduced a module on it for my classes. I thought it was insane to do three chapters on trigonometry, including solving useless equations with goniometric functions, and doing zero stats and probability. These students will likely (and statistically) choose uni careers such as medicine, psychology, law, politics -- the ones who choose engineering are very rare. Statistics is THE ONE field of math that they need.

ojosilva 3y ago

I would only teach applied math in high school, not the Applied Math curricula from most universities, but a vertical that covers things that can be made sense without depth, just breadth. I'd leave most abstractions and numeric solving for later in life or STEM students. I just feel I did so much equation solving for nothing in my student life! I remember too many times the teacher would assign the equation solving exercises 1 - 10 as homework, and only problems 11 and 12, leaving problems 13 to 20 out. Why? Well, problems are hard to teach, to grasp and to solve! Most math is actually very easy if kept easy, even integrals only have a few rules. Don't make a student do 700 variations of integral solving. Find a practical use for it (ie find the satellite velocity and position), introduce the subject with the problem "Today we're going to find a velocity of a satellite, there's a thing called integral that helps us do that, this is why and how..." and do that problem 20 times over and over. A variety of real-world problems have very simple math to solve, like stress and strain in materials engineering, center of gravity in physics, etc. Problem-solving is the worst of all math for most kids - so do repetitive problems and teach kids to extract information from problems into formulas and equations. And, yes, do plenty of statistics which is rich in real-world use too. Having a problem-solving, math applying mind and probabilistic view of the world are the skills to have at any field!

bluenose69 3y ago

One line of the article stood out for me: "College admissions officers value calculus, almost as a proxy for intelligence." I think what lack of Calculus signals is more lack of facility with Mathematics generally. When I speak with students who have weak (or missing) Calculus backgrounds, I usually find that they are uncomfortable with something much more basic: they simply do not understand what it means to use a symbol for a quantity. Not understanding that, they don't see how manipulating symbols (e.g. subtracting something from both sides of an equation to move a term to the other side) makes any sense. It's as though they had a mental block on the day when a teacher said "let x be the unknown". They usually bluffed their way through that class, and the next and the next, as they proceeded through middle school. But they never really got their heads around the ideas. And this, not intelligence, is the problem with their later success in STEM fields. Helping such students is a real challenge. It's a matter of establishing a connection and a trust that will let you probe back into their past until you find the place where the problem arose. This is like psychotherapy. It takes one-to-one work and it takes a long time to build trust, before the probing can begin. None of this is practical in a traditional college teaching framework, and that is why college admissions offers key on Calculus. As for the discussion of Statistics, I agree that this is more important for general students. STEM students need to add Calculus as well.

zozbot234 3y ago
> (e.g. subtracting something from both sides of an equation to move a term to the other side) 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.

OJFord 3y ago

How much time roughly do you spend in math classes in school in the US? I'm finding it a bit baffling reading these comments ('calculus vs statistics' etc.) from my UK perspective. We had I think roughly^ 3x1h maths lessons each week, and it didn't seem to be a problem to do both, as well as whatever else. Nor did we wait until secondary school to start stats (calculus sure, for the usual colloquial definition of it). Or are we talking specifically about some arbitrarily advanced level of each that somehow everyone's on the same page on without describing? (^it's been a while!)

wkjagt 3y ago

I’ve read a couple of texts by Paul Lockhart. His opinion (which I support) is that what is taught in high school as math isn’t math, as in what a mathematician does. As an example: Pythagoras proved the relation between the lengths of the sides of a right triangle. The journey towards finding the answer to this is math. Or even exploring why this relation even exists based on his findings is math. The answer to the question is expressed as a2+b2=c2. What we’re asking students to do is just to take that formula and apply it hundreds of times without going into the actual math behind it. This, he says, isn’t math at all. The whole “math” step is skipped. He compares it to if for example we were teaching music in school without ever touching a musical instrument, and just learning how to read and write notes. Or teaching art without ever touching a pencil or brush. The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. The same way we could totally miss out having students never discovering their artistic talents if art class was taught without ever actually doing any art. His text that explains this most clearly is “A Mathematician’s Lament”. It’s a great read and not very long.

j7ake 3y ago
Society thinks of math as technical and functional, rather than beautiful. 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.
- TeMPOraL 3y ago
  I disagree somewhat with both you and GP here: > Society thinks of math as technical and functional, rather than beautiful. The math people suck at is the technical and functional part, just as much as the beautiful parts. I'd frame the issue like this: there's the journey of arriving at the result, there's wielding the result, and then there's applying it to a very particular problem. Schools are teaching mostly the last part - applying a^2+b^2=c^2 to a set of contrived problems that are meant to exercise your ability in arithmetic, variable substitution, and simple symbolic transformation. They do not teach people how to wield the formula - as in, how can you realize on your own that any particular real-life challenge calls for application of that tool (or of mathematics in general). That's the middle part that's directly valuable, in the immediate term. That's the difference between a carpenter, and a person who figured out how to use a hammer to drive a nail into a wall. The beauty and journey part? That's study of history and art of inventing new types of hammers and other impactors, and perhaps inventing your own. Even further upstream of the "knowing how to use it to build something" part, but neither is taught in schools.

MarkusWandel 3y ago

Old fart here: In high school, math was broad spectrum. Calculus, Algebra, Geometry etc. Too early for specialized topics like Probability and Statistics. Those were first-year university material, where you start getting steered toward the actual math skills you'll need for your curriculum. What's wrong with that?

nameless912 3y ago

I felt this article deep in my old creaky bones. I hated math in middle school and high school. This is despite loving science, and enjoying math all the way through Algebra 1. However, when I started taking Algebra 2 and beyond, it seemed like we shifted away from why things work to solving ever larger polynomials and other tedious tasks, getting points taken off along the way for a flipped sign or digit or moving a decimal point, even when the entire rest of the calculation was correct. I ended up doing a degree in computer engineering, and every class that involved algorithm design or programming I excelled at, while every "hard math class" (for example, statistics, electromagnetism, etc.) I barely passed. I didn't fall back in love with math until I took discrete mathematics and signal processing in the same semester, and then later a class on algorithms and formal models of computation. They showed me that math was actually beautiful and fascinating, an art form unto itself. These three classes showed me how amazing math can be; I remember particularly that the day we learned about FFT's my brain felt like it had ascended into another dimension. They truly felt magical. Same thing with learning about recurrence relations and finally "getting" dynamic programming. Now I love math, and I am planning on taking this holiday break to finally crack open the Principia Mathematica and trying to really understand it (a personal goal of mine for several years). I can't wait to share the joy of mathematics with my kids when they're old enough. I just hope that they don't get bogged down in the mechanical, number crunching part like I did and manage to continue to see the beauty of the underlying ideas.

PM_me_your_math 3y ago

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bikenaga 3y ago

Some comments about points people have raised ... 1. Can you teach statistics without doing calculus first? It's quite common. We had such a course which was typically taken by social science majors who needed some statistics in their fields. (I never taught it.) The prereq was just a little algebra. A typical text was Mario Triola's "Elementary Statistics" (see here for the table of contents: https://www.amazon.com/Elementary-Statistics-13th-Mario-Triola/dp/0134462459 https://www.amazon.com/Elementary-Statistics-13th-Mario-Trio...). The students used Minitab as part of the course. But we also had prob & stat courses which used calculus. 1. (a) How can you define the exponential function (or the number e) without calculus (limits)? When I taught college algebra (= review of high school algebra, around two courses below Calc 1) I'd say something like: "There's a number called e which is approximately 2.7118281828459045 ... - it's kind of like pi, an infinite decimal that never repeats. We've seen there are exponential functions like 2^x, and we have an important exponential function called e^x. You're probably wondering why we're using such a weird number in an exponential function. You'll see how it comes up if you go on to take calculus." No one much complained about any of that. Even a typical calc course skips a lot of the theoretical background - you don't see the background unless you take a course in real analysis. In teaching college math, you're always "starting in the middle", assuming a lot, and in basic courses you're omitting a great deal of the rigor. (Also, in calculus it's common to define logs first using a definite integral, then discuss inverse functions, then define e^x as the inverse of ln x [so in particular e is just e^1]. You justify the notation by showing ln x and e^x have the properties you "expect" logs and exponentials to have [e.g. ln (a b) = ln a + ln b].) 2. Why not teach statistics/linear algebra/logic in high school? High school curricula are often strongly determined by state testing, and by the needs of students who are going on to college to have the math that colleges expect. I took courses in math logic and matrix theory in high school, but they were electives; the "big" course for me from the point of view of college admissions was BC calc. I certainly got no college credit for the logic or matrix theory courses. So e.g. if you tried to do linear algebra or logic in high school it would come at the expense of topics that a state is testing - then your students do poorly (even though they might know some pretty good math) and you and your school get in trouble. There is more of a movement to teach stat and data analysis in high school and college - things are changing. 3. Why not teach statistics/linear algebra/logic before/in place of calculus in college? College math departments are heavily service departments. The bulk of the credit hours are in courses taken by majors from other departments. Those departments (science, engineering, business, and so on) tell math departments what their majors need, and math department have to ensure that the math courses suit. The other departments in turn are often required to have majors take certain courses with certain topics by program accrediting agencies. So there can be a lot of inertia there.