I can't shake this feeling that we're just teaching maths wrong? I keep coming back to this Numberphile video which compares maths to art class, that we don't try to inspire maths in the same way....
I can't shake this feeling that we're just teaching maths wrong? I keep coming back to this Numberphile video which compares maths to art class, that we don't try to inspire maths in the same way.
I don't necessarily buy the argument of maths having a higher barrier to entry compared to great art or literature either, done right I think it's possible to get a curious person interested in mathematics without scary formulas and concepts.
we are obviously teaching everything wrong i don't know what is in that video, but classic wrt maths is mathematician's lament https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician's_Lament.pdf
I've never understood why all the cool things that you can do with math aren't part of how it's taught. All the math problems when I was a kid were about divying up ridiculous amounts of produce...
I've never understood why all the cool things that you can do with math aren't part of how it's taught. All the math problems when I was a kid were about divying up ridiculous amounts of produce or measuring the heights of random buildings that no kid in that class could possibly care about. I know the mathematical concepts needed to build things or understand a lot of the actually fun things are too advanced for most kids, but it's not impossible to explain how those things are going to be used as building blocks. Heck, make the kids read the mathiest days from XKCD Explained and use that as a springboard.
I'm not saying it's easy, but at least when I was a kid there seemed to be extremely little realistic effort to make math interesting or enjoyable to kids. Not zero effort, there were those word problems, but none of them felt like any thought had gone into what kids might want to use math for.
I only have experience teaching grade 8 algebra with one kid, so survivorship bias and extremely unreliable sample size warning. The kid gets frustrated with tiresome busy work very quickly, as...
I only have experience teaching grade 8 algebra with one kid, so survivorship bias and extremely unreliable sample size warning.
The kid gets frustrated with tiresome busy work very quickly, as would we all. So the curriculum has to rotate out types of problems and mix new with varying kinds of old.
Humour helps: make the word problems or situations extremely silly, such as a dog wants to sniff X number of butts or I want to stack Y number of skittles on the cat. Make them completely absurd. Your point on XKCD and explained exactly. So many of Randall's videos of what ifs solve absurd problems using a lot of maths for just plain silly fun.
Usefulness is also underrated. Baking with imperial units of measurement, scaling recipes up or down, min maximizing the quantity of sugar cereal you can buy by calculating the mass per dollar, EV training your pokemon, watching savings account interest NOT grow perceptibly against inflation vs equity...
You can't build a skyscraper without a foundation, so there's going to be boring bits, but get a good balance of old/new/humour/relevance in there and it's very doable. The trouble with public education of maths has always been and will continue to be trying to find that balance, but now with 30 kids all at different levels of foundation and feelings and interests.
Parents have to retake the reins on this one I think. School has been proven to be terrible at teaching maths. You gotta do it yourself
Parents who mostly learned math in those same schools except they haven't brushed up on any of it in decades? This is a ridiculous ask. I'm the sort of person who watches Numberphile and has an...
Parents have to retake the reins on this one I think. School has been proven to be terrible at teaching maths. You gotta do it yourself
Parents who mostly learned math in those same schools except they haven't brushed up on any of it in decades? This is a ridiculous ask. I'm the sort of person who watches Numberphile and has an M.Sc, and I'd still be pretty bad at teaching a kid math. And most parents have less math literacy and less time than I do.
I'll tack onto this: I took a class in college about teaching math to elementary school students. The instructor was a very passionate man about education and math, one of the best teachers I've...
I'll tack onto this: I took a class in college about teaching math to elementary school students. The instructor was a very passionate man about education and math, one of the best teachers I've ever had. Half the reason I took that class is because I had him for my first-semester math class, and he made it so much easier and more enjoyable than any other math class I've ever taken.
He told us an anecdote about having to sit in on an exam for another teacher once, and he decided to glance at the exam while the students worked on it. Even though he had taught that subject in the past, he couldn't remember how to solve any of the problems because it had been a few years since he'd taught it.
If a deficated and passionate math teacher can forget how to solve certain problems after a few years, I wouldn't expect parents to remember after 20+ years.
I'm at a loss myself, in all honesty, on how to do it myself, if by "it" we mean how to teach math properly and how to become a numberphile and see beauty in it. All I can aim for is a healthy...
I'm at a loss myself, in all honesty, on how to do it myself, if by "it" we mean how to teach math properly and how to become a numberphile and see beauty in it.
All I can aim for is a healthy happy child who can push through the ugly slog, coming out alive and competent enough to get their degree and never look at math again.
I'm definitely not claiming that I can be the sort of mathematician that The Lament wants, someone to inspire love and wonder and appreciation of math. All I can do is monitor individual attention, make daily adjustments, try to be firm but flexible, encouraging without coddling, and then requiring that the kid put in their hours doing the boring, soul crushing, meaningless and ugly math that society is requiring of them.
I hate it, I wish it wasnt this way, but we will be able to push through and so have millions of parents. Math has to stop being presented as a sort of elective because this sick and disgusting society isn't treating it this way.
My parents never graduated elementary school and have never done algebra, but they paid for tutoring to make sure we can do it. There's a weird American culture that drop off kids at school and pretend that's all parents need to educate: this attitude has to change.
No one has to love math: this is a tragedy. But also parents need to stop expecting the school will do it either. Mr Incredible spent a lot of time sitting with his kid doing "new math" -- it isn't the curriculum, it's spending the hours doing the ugly boring irrelevant math together. Seeing your parent struggle, complain together, but persisting -- that, I believe, is more realistic and more helpful than asking more of our teachers who are already tasked beyond the impossible
Having the time to tutor your kid yourself or the money to hire someone else to do it is a privilege, though, and as a result relying on parents to teach their kids math only reinforces existing...
Having the time to tutor your kid yourself or the money to hire someone else to do it is a privilege, though, and as a result relying on parents to teach their kids math only reinforces existing inequalities. The entire point of free public schools is that students can be educated to the same standards (at least ideally) even when their parents have vastly different resources. The way public schooling is funded in the US already undermines this goal, since schools are mostly funded based on the wealth in their districts, but relying on parents rather than teachers only exacerbates that same issue. Deciding to teach your own kid math at home is perfectly reasonable as a choice to optimize things for them specifically as an individual, but making that the general solution upon which we base decisions and public policy is bad for society.
You insist that asking teachers to do more when they're already asked to do the impossible is not viable, but the reason that it's impossible is because there are far too few teachers and they're paid a pittance. Those are both problems that could be solved through actually properly funding schools and paying teachers a wage that even kind of approaches their societal value. You can argue that's unrealistic in our current political climate, but it's no more unrealistic than expecting kids to get math tutoring outside of school when their parents are working multiple jobs to make ends meet while living paycheck to paycheck. Unless you fundamentally don't care about kids having grossly different opportunities and education based on their parents' income, it's actually a lot more feasible to pay teachers better than it is to ensure all parents are capable of teaching their children math at home.
100% in agreement with you. I wish funding education was the priority that it should, and teaching wages was the focus and lion's share of these resources, not buying more laptops or building more...
100% in agreement with you. I wish funding education was the priority that it should, and teaching wages was the focus and lion's share of these resources, not buying more laptops or building more gyms or whatever "hardware" "feels" more valuable than just - plain - wages - for - good - people.
I might sound like I'm contradicting myself : speaking as a citizen of how we should improve a system often comes in conflict with how I should parent my one child here an now who is only a child for x number of years. What I think society should do is often diametrically opposed to advice I would give to actual on the ground parents who don't have years/decades/votes to change the system -- I think it's evil that we're having to choose between voting with our kids' limited time in public vs private / tutor, and how these things then get funded.
It's like asking for people to stop driving and take transit, because if no one takes transit it won't get funded, but no one can take transit right now because it's terrible. I think these are intentionally manufactured problems and politically I will always vote / volunteer to change them, even as I live as part of the problem. =..= very depressing...
Just to affirm what you’re saying, having been a former Kaplan test prep instructor, my takeaway was that the main value add of those courses has less to do with the curriculum and more the amount...
The kid gets frustrated with tiresome busy work very quickly, as would we all. So the curriculum has to rotate out types of problems and mix new with varying kinds of old.
Just to affirm what you’re saying, having been a former Kaplan test prep instructor, my takeaway was that the main value add of those courses has less to do with the curriculum and more the amount of practice they force students to do. The practice tests and problems build up the mental and physical stamina to be able to sit still for multiple hours to grind through the exam. Most people who did fine in school ought to know the majority of the content, it’s just that they tire out. Especially on the reading comprehension and problem solving elements that require focus.
We expect everything we learn to be fun and inspiring, but sometimes you just have to clear a hurdle by getting a lot of reps in through rote memorization. We don’t doubt this when we talk about fun things. If you want to get good at basketball you practice free throws to develop a sense for how to throw the ball. If you want to get good at piano you have to practice until you can reach the notes instinctively. If you want to get good at video games you play enough to where you know the layout of the map and can have fast enough twitch reflexes. And if you want to be good at math it sure helps to just be able to know, reflexively, that 9x9 is 81 without needing to stop and check or calculate. Getting the mental calculation smooth enough to where it doesn’t introduce friction into your reasoning process is what actually lets you unlock more of your brain’s capacity to do the more abstract reasoning you need for algebra and calculus.
Parents have to retake the reins on this one I think. School has been proven to be terrible at teaching maths. You gotta do it yourself
Part of the problem is that most parents are also terrible at math. And I think a lot of people have a sort of mental block with it. Being Indian I notice a similar effect with my name. I usually just give my initials at coffee places to avoid mispronunciation and more than half the time people ask me “How do I spell it?” The brains are on autopilot to where they see an Indian person and just assume whatever comes out of my mouth when they ask my name is going to be unpronounceable so they try to skip ahead and don’t processing that I just told them two letters. A lot of people do this with any kind of arithmetic where they just see numbers of any kind (and God forbid those numbers be mixed with symbols) and their brains check out.
Bolded my change. While what you said is correct, I have met far too many elementary teachers (or aspiring teachers in my college years) with math anxiety. Math is not some insurmountable tricky...
Part of the problem is that most elementary teachers are also terrible at math. And I think a lot of people have a sort of mental block with it.
Bolded my change. While what you said is correct, I have met far too many elementary teachers (or aspiring teachers in my college years) with math anxiety. Math is not some insurmountable tricky magic; it's just math.
I'd also add that much of what makes students hate math is arithmetic. They get bored and frustrated, so they check out, thus causing gaps in their knowledge that make later topics actually insurmountably challenging. Math is unique in how rigid its prerequisite structure is—there is no need to start the study of history with the speciation of H. sapiens.
yes! Exactly! Both that (1) there's a "muscle" to the very fundamentals, and (2) cultural mental block There's a ditch on either side of the road: kids will kill themselves if pushed too hard, but...
yes! Exactly! Both that (1) there's a "muscle" to the very fundamentals, and (2) cultural mental block
There's a ditch on either side of the road: kids will kill themselves if pushed too hard, but kids will also default to not ever automatically knowing 9x9=81 if they didn't spend enough time for their brain to register it into longer term memory.
I greatly enjoyed Mathematician's Lament, but at the same time I reflect back on my years spent in Canadian high school art and band classes, and how much of a waste they were. I was never taught colour theory, art history, circle of fifths, transposition or any of the boring bits. Everything was just a circle-praise of "wow, I love this, A+" and be free and here's more empty canvas! I did have fun, I appreciate some types of art and music, but I'm absolutely ruined for dedication and concentration and effort towards making good music and good art: "everything is music everything is art hurray!" Thank goodness I didn't need art and music to obtain a level of life where I, now as an adult, can choose to do more music and art out of pocket. So, credit: I came out not hating art/music, I love it.
Could the same thing happen? Do citizens who have a grand and lovely time with math in high school go on to enjoy math and participating in the thinking aspects beyond passively watching a video mathematician enthusiastically talk about math? I highly doubt it. It's okay to push a little bit, early and often and kindly.
Question: what were your Kaplan kids like? What about the kids whose parents didn't send them to tutors / who made such a fuss they withdrew -- could you teach them the same way or how do we teach them?
I was teaching the MCAT and I was, like, 2 years older than my students so it’s not the most typical teacher/student relationship. I also only had one cohort of students. It’s a medical school...
I was teaching the MCAT and I was, like, 2 years older than my students so it’s not the most typical teacher/student relationship. I also only had one cohort of students. It’s a medical school admissions test so they tended towards being highly motivated gunners who had different levels of heart in it. Everyone was motivated to do well on the test for that cohort so the discipline was mostly there, but there was a decent number whose heart just wasn’t in it. They were already burned out before even being done with their bachelors degree so you can tell that even if they’re smart enough to score well they’re not gonna make it all the way through the slog that is medical school.
Hmmm interesting sample...... highly motivated young adults who chose this program with the math requirements is a very different circus than 8th graders who gain social benefits from complaining...
Hmmm interesting sample...... highly motivated young adults who chose this program with the math requirements is a very different circus than 8th graders who gain social benefits from complaining about math and rebelling against being told to do things in school.
I don't know that parents can be expected to have the resources to do that. Teaching these concepts well, in an interesting way, takes a much more multifaceted and nuanced skillset than teaching...
I don't know that parents can be expected to have the resources to do that. Teaching these concepts well, in an interesting way, takes a much more multifaceted and nuanced skillset than teaching them in a boring, frustrating way, and that's assuming that they have grasp of the basic math to teach them these concepts at all.
One of the benefits of a school system should be the variety of disciplines that can be pulled from to work together in the creation of curricula like this. Obviously teachers aren't given the resources required to pull this off, or the respect they deserve when they manage to do it, I am absolutely not blaming teachers en masse for this. Parents don't have all of those resources to pull from, and they have other jobs on top of it.
I know it's not realistic to expect this from our public school systems, but it's also not unreasonable.
I completely agree that not all parents have, frankly, this luxury to either do it themselves or hire a tutor who could. Then we circle back to an equity issue, and if we then multiply it by the...
I completely agree that not all parents have, frankly, this luxury to either do it themselves or hire a tutor who could. Then we circle back to an equity issue, and if we then multiply it by the horror of requiring calculus / discreet maths for a high paying job, essentially we have an all in one package to perpetuate cycle of poverty, blaming the citizens for choosing to not try hard enough at math.
I did grow up with the East Asian system of pushing super hard: at least two hours a day [edit: 6 days a week plus at least 1 hr home work a day for just math] for kids who get it, mandatory afterschool tutoring for kids who don't, and "optional" afterschool tutoring for kids whose parents will require post sec education of them. The down sides are laid out in @Moonchild 's mathematicians lament. But that super long (excellent) essay is forgetting the "up sides" (shudder) of pushing far harder than Americans systems are prepared to do.
Strong air quotes around up sides. Very strong. Kids who don't kill themselves, literally, will be able to do algebra and post sec math and get good jobs. All those college entry boards know this: they see far more well pushed kids than kids who "are interested" in math naturally.
Fact is. Society is sick, and has thrown up these hurdles between kids and a financially secure future. We can either change society or we can bend our kids to fit into it. Some kids will break instead of bend, but most kids will bend with enough pressure. What this one states is doing is wasted effort; it's pushing without pushing enough, making everyone miserable without fitting kids into the mandated slots. It's like bad braces or starving followed by binge.
That sounds horrible. I’m glad I was able to enjoy math within the system. I want to give credit to the handful of math teachers at my public high school that were excellent. My statistics teacher...
That sounds horrible. I’m glad I was able to enjoy math within the system. I want to give credit to the handful of math teachers at my public high school that were excellent. My statistics teacher loved the subject and teaching so much she’d occasionally jump up and down with excitement during a lesson. My calculus teacher was one of the kids’ favorite in the whole school. He managed to make it fun while caring about each student and explaining things well.
In HK, I got the ruler a lot. On a good day it's smacks to the palm, not the back. Social shame was a popular one. All kinds of schoolyard and domestic abuse that hopefully don't happen anymore....
In HK, I got the ruler a lot. On a good day it's smacks to the palm, not the back. Social shame was a popular one. All kinds of schoolyard and domestic abuse that hopefully don't happen anymore.
In a Canadian Highschool - I've had a few really excellent math teachers as well. One of them has a (small) roster of repeated dad jokes and sometimes tries to freehand draw the perfect 1m+ diameter circle on the white board -- the lameness of jokes works GREAT on us to collectively groan in solidarity, and we just LOVE to nit pick where a circle might not be perfect. Another was just a great dad type figure, so we respect him when he says "alright, lets' get back to X". Another one is a jolly woman called us her little chickadees and has an infectious laugh. Attitude and excitement are infectious, indeed. Kindness helps. Patience definitely.
But even those wizards wouldn't have been able to do much with a class full of kids who are just marking their time, singing and spacing out and talking, like the 8th graders in the article. If the prevailing culture climate is "math is hard/dumb/useless - pass all the ones you can, go to Track A if not, and beg for C+ until they're no longer compulsory", kids won't put in the mental work to concentrate enough to actually benefit from their kind wizardry.
The braces thing is a really good analogy. I think a lot of people just want things to suck for suckiness’ sake, like it will forge them into being stronger. But the stress is a cost you pay to be...
What this one states is doing is wasted effort; it's pushing without pushing enough, making everyone miserable without fitting kids into the mandated slots. It's like bad braces or starving followed by binge.
The braces thing is a really good analogy. I think a lot of people just want things to suck for suckiness’ sake, like it will forge them into being stronger. But the stress is a cost you pay to be able to get through, it’s not the thing of value in itself.
On the point of usefulness, something that I think gets ignored by curriculums far too often is that with how some peoples’ minds work (kids and adults alike), for subjects that the individual...
On the point of usefulness, something that I think gets ignored by curriculums far too often is that with how some peoples’ minds work (kids and adults alike), for subjects that the individual doesn’t find inherently interesting, usefulness isn’t just a nice-to-have, it’s nearly a requirement.
I’m like this. The difference between how deeply and quickly I learn something that I need to know in order to accomplish a task vs. something that feels arbitrarily foisted upon me by a course with no immediate purpose other than to pass an exam is staggering. As an adult I have the patience and discipline to slog through the latter (though I’ll still hate it), but in middle/high school and uni it was a very different story. Naturally, this made all maths from algebra up a struggle for me, despite having a good enough grasp on logic to wind up working as a programmer.
Schools all the way from elementary up through high school don’t do anywhere near enough hands-on to make practical use of learnings. Uni is a bit better but still has plenty of room for improvement there.
agreed. In one episode of Clarkson's Farm, Caleb admits to having been absolutely terrible at math. But when he approaches a problem in the form of farming, a vocation he loves, he was able to...
agreed.
In one episode of Clarkson's Farm, Caleb admits to having been absolutely terrible at math. But when he approaches a problem in the form of farming, a vocation he loves, he was able to work out nearly instantly how much money could be made if x grain was purchased at y price and planted in z acres with harvest rate of a% sold at b dollars per c quantity. If school had been like this for Caleb, he would have absolutely been an A maths student.
In my experience as a student anyone with a desire to teach mathematics has also a strong desire to avoid anything soft and squishy like emotion and inspiration. It might be easier to train art...
In my experience as a student anyone with a desire to teach mathematics has also a strong desire to avoid anything soft and squishy like emotion and inspiration. It might be easier to train art majors to teach maths than to persuade a "real" math teacher to care about inspiring anyone who is not in love with math already.
I've definitely been fortunate that many of my maths/science teachers weren't jaded and boring, but nonetheless I think mathematicians have a deep love of "squishy" things like beauty and comedy....
I've definitely been fortunate that many of my maths/science teachers weren't jaded and boring, but nonetheless I think mathematicians have a deep love of "squishy" things like beauty and comedy.
They talk frequently about equations or proofs being beautiful (e.g. Euler's identity), and the writers for Simpsons/Futurama are accomplished mathematicians.
I would propose what you experienced was people who loved math and were required to teach. Students regularly had this idea that teaching was what I spent my time doing when in reality it was a...
I would propose what you experienced was people who loved math and were required to teach.
Students regularly had this idea that teaching was what I spent my time doing when in reality it was a handful of hours a week. I liked teaching but many of my peers did not.
That is what I thought of teachers coming from engineering or physics, but some of the worst math teachers I had were actual mathematics majors with licenciature (meaning they:already majored as...
That is what I thought of teachers coming from engineering or physics, but some of the worst math teachers I had were actual mathematics majors with licenciature (meaning they:already majored as educators) or "magistério" (a course you can take in Brazil to become a teacher).
They where all so bad. At least Humanities and Arts teachers showed passion sometimes. Math teachers were dead inside lol
Math is a subject that requires deep work to master, lending itself to individual and small group formats. Reading, re-reading, applying in practice, watching examples, and formulating questions...
Math is a subject that requires deep work to master, lending itself to individual and small group formats. Reading, re-reading, applying in practice, watching examples, and formulating questions is necessary as you get into calculus.
While I believe that every person is capable of developing these habits and applying them to math, not everyone is ready at the same time. Kids in particular develop in different ways at different ages. Until someone is developed enough to have the ability to work individually to some extent, you really can't force understanding on them. There's no amount of rote demonstration that will convey understanding of epsilon-delta proofs, for example, and those are week 2 or 3 of a calculus course (if you are teaching for material mastery). So if your goal is more people who take calculus successfully, you need a mechanism to assess when they are ready to start learning the concepts that are beyond the "math facts" variety.
While having an engaging curriculum is helpful, there is no set of exercises that will get a student to a point of proficiency without them having some ability to study and learn. So we should focus on making sure we teach kids how to learn, and not lock them out of opportunities just because they developed those skills a little later than their peers. More emphasis on competency based placement in K-12 courses rather than age-based might help, but it is a difficult problem to fix within the confines of our schooling system and the lack of resources many students have at home.
This is one of the reasons I like the Montessori method for teaching. Classes combine students from three years together. At my daughter's current school, this means 3yo-kindergarten, 1st-3rd...
While I believe that every person is capable of developing these habits and applying them to math, not everyone is ready at the same time.
This is one of the reasons I like the Montessori method for teaching. Classes combine students from three years together. At my daughter's current school, this means 3yo-kindergarten, 1st-3rd grade, and 4th-6th grade.
This makes room for the "sensitive periods" so that when interest sparks on something, the student can deep dive into it. But an equally important piece is that the younger kids see what the older kids are doing, so they have a sense of where they are headed. As they move up, they become the leaders in the classroom and often teach the younger kids, which provides reinforcement.
The work is also integrated so that math becomes part of other subjects, which allows them to apply what they have learned in practical ways.
My daughter is only in the fourth grade, and I only did Montessori through second grade.
So I don't know how well the methods translate into high school subjects like algebra and calculus or the more categorically focused sciences (biology, chemistry, physics). There's no Montessori high school available locally anyway.
My hope is that she will have enough foundational knowledge, confidence in her ability, and willingness to ask for help that she can excel in a more conventional environment for middle (generally 7-8th grade locally) and high school (9th-12th grade).
One thing that's difficult applying these lessons to conventional schools (speaking from a US perspective) is that the structure is radically different, so a single teacher's ability to implement these principles is limited, both because they have little control over their curriculum and because they need to prepare students for a specific next class. It also requires different training and a different mindset from the teachers, though from my observation, it's more fulfilling for the teachers overall.
The specific topic of epsilon-delta proofs may be one of the few topics that aren't an actual blockage for students who don't get it in the allocated curriculum time. They're necessary so students...
There's no amount of rote demonstration that will convey understanding of epsilon-delta proofs, for example, and those are week 2 or 3 of a calculus course (if you are teaching for material mastery). So if your goal is more people who take calculus successfully, you need a mechanism to assess when they are ready to start learning the concepts that are beyond the "math facts" variety.
The specific topic of epsilon-delta proofs may be one of the few topics that aren't an actual blockage for students who don't get it in the allocated curriculum time. They're necessary so students don't take the theorems and derivative formulas by faith, but not understanding them doesn't preclude understanding the application of limits and integrals.
Contrast with most of the rest of math, where there is a strong sequence that must be followed to build understanding.
Are epsilon-delta proofs just a fancy way of saying "learning how limits work"? I took calc 1 and did very well, but I have no idea what an epsilon-delta proof is. But at the stage of calc 1...
Are epsilon-delta proofs just a fancy way of saying "learning how limits work"? I took calc 1 and did very well, but I have no idea what an epsilon-delta proof is. But at the stage of calc 1 they're referring to, we were learning how to work with limits, which I recall making the subsequent move to derivatives seem very intuitive, so I'm wondering if the same material was covered without using that terminology.
You are correct. Epsilon Delta proofs are the mechanics of limits. They are often touched on at the start of calculus one, and only re-covered in the first semester of real analysis which is...
You are correct. Epsilon Delta proofs are the mechanics of limits. They are often touched on at the start of calculus one, and only re-covered in the first semester of real analysis which is largely limited to math majors and some smattering of STEM folks who go hard on the theory side of things. They are important to cover conceptually, but unwieldy to apply.
Edit: they also notoriously give students fits as the first and sometimes only exposure to thinking in terms of proofs.
I don't recall whether I actually learned any proofs specifically when I learned limits in calc one. I definitely learned the intuitive concept of limits, but it was over 10 years ago so I may...
I don't recall whether I actually learned any proofs specifically when I learned limits in calc one. I definitely learned the intuitive concept of limits, but it was over 10 years ago so I may just not remember the proofs side that clearly. We definitely didn't do epsilon-delta proofs in the refresher course I took in the first semester of my M.Sc, but that course was trying to combine calc 1, basic stats, and linear algebra all into one semester so we could handle machine learning in the spring, so everything was at least a little rushed.
It's a tough curriculum choice to make. Many college level courses for stem degrees will cover them in a lecture, maybe have some problems on the homework, but not test them. Other programs might...
It's a tough curriculum choice to make. Many college level courses for stem degrees will cover them in a lecture, maybe have some problems on the homework, but not test them. Other programs might skip them entirely, especially if they don't have separate stem and non-stem calculus courses.
Personally I like teaching them because if nothing else it exposed students to a type of thinking they might not otherwise see. Putting them on a homework, but not testing seems like a decent compromise.
The downside is that math programs tend to push more and more of the "real" math to high level courses, which can create a fourth year weed out of real analysis, which is a harsh situation for math majors to have.
Edit: I don't think they are covered in AP calculus, but I could be misremembering. Most of the students I spoke to didn't see them until college even if they did AP calculus.
I took calculus in a post-secondary enrollment program (where I took classes at a local college as a highschool student) rather than taking AP courses, so it was dependent on the whims of my...
I took calculus in a post-secondary enrollment program (where I took classes at a local college as a highschool student) rather than taking AP courses, so it was dependent on the whims of my specific math professor essentially (and I learned a lot from that class and it was the first math course I'd ever taken I didn't hate, so kudos to her for that.)
It's always great to hear about good teachers. I used to dislike math until I had a great calculus professor, and that led me to getting degrees in math. It's amazing what a good presentation of...
It's always great to hear about good teachers. I used to dislike math until I had a great calculus professor, and that led me to getting degrees in math. It's amazing what a good presentation of the material can do.
One of my regrets is that I let my dad convince me to take accounting the subsequent semester instead of trying out number theory like that prof suggested. I think that might've led to me doing...
One of my regrets is that I let my dad convince me to take accounting the subsequent semester instead of trying out number theory like that prof suggested. I think that might've led to me doing more real math in college if I'd found it interesting, whereas accounting was incredibly easy but equally boring. But I ended up doing linguistics and focusing on formal semantics, which involves a bunch of set theory and lambda calculus, so I at least got something math-related in there eventually.
I picked those proofs because they defy rote memorization, like most proofs. But I would say that most of the topics in calculus can't really be absorbed or mastered through rote memorization....
I picked those proofs because they defy rote memorization, like most proofs. But I would say that most of the topics in calculus can't really be absorbed or mastered through rote memorization. Even things as simple as the application of the chain rule takes an understanding of the process in order to apply to arbitrary problems. I suppose you could memorize algorithmic steps for much of calculus, but suspect you would get destroyed on any real test of the material that asks you to apply calculus to a situation. E.g., using it on the context of physics or other sciences.
Developing intuition for rates of change, rates of rates of change, inflection points, families of solutions, etc. are important to any real mastery of the material. And those are the things that can't really be rushed in the K-12 space. Kids need to have developed the healthy mental habits to foster learning in order to really learn these topics.
My teachers called it number sense. In a less advanced situation, it’s the gut instincts that tell you when you miskeyed the arithmetic in your calculator.
My teachers called it number sense. In a less advanced situation, it’s the gut instincts that tell you when you miskeyed the arithmetic in your calculator.
I never heard it called that, but I had some teachers teach me exactly this. First, does the answer make sense in the context of the question? If you calculate the moles of a substance and get a...
I never heard it called that, but I had some teachers teach me exactly this. First, does the answer make sense in the context of the question? If you calculate the moles of a substance and get a negative number, you did something wrong. Then, you can do the problem in your head considering only the magnitude. If your mental math says you should be around 8 orders of magnitude and your answer is 3689, you definitely did something wrong. I am sure there are other tricks, but those two saved me a lot of wrong answers. Also the magnitude only calculations are surprisingly useful in daily life.
I learned integration by parts this way in high school, and indeed I was destroyed by it on tests. I (and most of my peers from what I recall) did our best to memorize the algorithm and try to...
I suppose you could memorize algorithmic steps for much of calculus, but suspect you would get destroyed on any real test of the material that asks you to apply calculus to a situation.
I learned integration by parts this way in high school, and indeed I was destroyed by it on tests. I (and most of my peers from what I recall) did our best to memorize the algorithm and try to recognize when/where to apply it, but I always found that recognition impossible. Differential equations in undergrad was awful for similar reasons, being handed some list of divine algorithms and told to, somehow, recognize which one to use in a given situation. Again, I was destroyed. My only D in undergrad, but hey, you can graduate with a few D's and life moves on.
It is not until graduate school that a professor makes an offhanded comment that we'll need integration by parts, and that's just product rule. Earth-shattering revelation; integration by parts feels easy now. I really wish someone had said so in high school, but instead the focus was on this rote memorization of algorithms. Or, if someone did say so, I wish I had paid attention. I often wonder if I was told things like that, but I just lacked experience or intuition at the time to recognize the value of it.
It's amazing what the right presentation of a method or idea can do for understanding. Differential equations is especially notorious for crushing students and forcing desperate memorization,...
It's amazing what the right presentation of a method or idea can do for understanding. Differential equations is especially notorious for crushing students and forcing desperate memorization, though I blame some of that on overly full curriculums.
I have another thesis: It's not age, it's cumulative hours spent on maths prior to algebra. What this state is doing seems to be offering algebra by 8th grade. This created an awkward gap between...
Not all eighth graders are ready for the abstract concepts — like variables, linear functions, slope — that come with Algebra I, some experts and teachers say. [...]
I have another thesis: It's not age, it's cumulative hours spent on maths prior to algebra.
“This whole idea is a very naive belief that if we just kind of make it for everybody, everyone will learn,”
What this state is doing seems to be offering algebra by 8th grade. This created an awkward gap between too early and way too lax. You don't just offer maths, you have to make it compulsory early and often so that being able to do basic pre algebra maths is an achievable requirement. If kids are already struggling with understanding scientific notation or don't have their multiplication and division effortlessly, that's too early. But by the time they're 13+ they're so far behind that doing the calc route becomes nearly impossible to catch up to kids who put in the cumulative hours in their primary years.
She got As in elementary school math but her grades fell once she hit algebra in eighth grade. “I was just really immature and didn’t pay attention,” she said. “And I needed more time — some people learn slower and others faster.”
Elementary school maths is laughable. What are they doing. I know they want to discourage rout learning and memorizing, and they want to emphasize higher reasoning etc. but at some point it's like trying to build a car using hand fashioned tools made of clay: there's so little time to develop the automated portions that the basic tools required to quickly and painlessly do algebra geometry calculus just aren't there.
“Then you started adding these symbols, and I didn’t get it.” But he’s confident he’ll master it. “It just takes time,” he said.
It takes time. Time the primary years should have accounted for. The tragedy here isnt pushing algebra, it's not enough pushing prior to algebra.
What the district really needs to address its math gap, he and other administrators said, are more certified math teachers, math tutors to help those struggling and smaller class sizes.
You can't get there from couple hours of easy loosely goosey fun maths a week. You gotta have smaller class and individual tutoring and sheer time spent doing the basics. Judging by how few American adults choose math, the few who do move onto tech jobs -- who would want to teach math when it barely pays enough to live / student loans. It's a complex problem decades in the making.
If that's the biggest barrier I guess the entire education system is in danger. And always has been for as long as we had structured education. But I digress. It's an interesting idea, but...
“Eighth grade, they’re just in full-on puberty, hormones,” said Zach Loy, another math teacher at the high school, an hour’s drive from Minneapolis. “Are they capable of sitting down and focusing on one thing for two, three minutes at a time without getting distracted? I see that as being the hardest barrier.”
If that's the biggest barrier I guess the entire education system is in danger. And always has been for as long as we had structured education. But I digress.
It's an interesting idea, but mathematics is an area where fundamentals are everything. if you're 1st-7th graded mathematics aren't rock solid, you're going only see an amplification of struggles for algebra. I'm sure there's a myriad of factors that can improve in this regard, but it ultimately comes down to each individual everyone has different proficiencies, and rushing them is the last way to help here.
For reference, I did do an accelerated track and I'll admit that I struggled a bit in the hybrid Algebra II/Calculus A course. You know what helped the most in Calc B/C? The teacher took 2 weeks to reinforce our trigonometry . I can't really explain what "clicked" that 2nd time, but seeing trig again after that exposure to how growth and limits work kinda just did something. I suppose the first time through I was relying a lot on rote memorization (and tbf, a good part of trig is just accepting seemingly random identities). The 2nd time I could understand more of the why's and how's behind how the curves are modeled, and received more pratical applications of what they are used for besides triangles.
I'm pretty sure I was taught these concepts in eighth grade, back in the day (in Portugal). It was one of the only two classes I ever had a negative grade in (the other being 5th grade English),...
I'm pretty sure I was taught these concepts in eighth grade, back in the day (in Portugal). It was one of the only two classes I ever had a negative grade in (the other being 5th grade English), and the teacher was pretty decent. Eventually he gave me some personal tutoring and I got past that hurdle, then it was smooth sailing all the way to the end of high school (calculus in high school here ended with basic limits and derivatives).
I'd say these lessons can be tough to tackle for the first time in your life but they're so important and fundamental that I'm absolutely behind their early introduction. IIRC most of my classmates got through it, and even an idiot like me! No phones or laptops at the time, though (this definitely stands out in the article).
Tangential: Saw this math problem on Reddit a few days ago: the answer is .... ? Right? 90mph on the return trip right? Most everyone says it's impossible and you'd have to travel at the speed of...
A traveler must make a 60-mile round trip between two towns, Aliceville and Bobtown. The distance each way is 30 miles. Going from Aliceville to Bobtown, the traveler drives at exactly 30 miles per hour. By the time they reach Bobtown, they decide they want to average 60 miles per hour for the entire 60-mile journey.
Question: How fast must they drive on the return trip from Bobtown to Aliceville to achieve an overall average of 60 mph?
the answer is .... ? Right?
90mph on the return trip right?
Most everyone says it's impossible and you'd have to travel at the speed of light and still not increase the velocity to an average of 60 mph.
If folks are wrong, how are this many people wrong? If they're right can someone explain this to me? One possibility is that people (potentially myself included? ) got trained by school math to pick a formula, then plunk numbers in and viola. There must be some common pit trap here where people are trained to automatically picking a bad formula to begin with with? Maybe it has to do with mph being a rate and adding and averaging rates is less intuitive?
The pit trap is that total average speed is not the average of speeds. If you're trained to pick a formula, you probably see "average" and pick (a+b)/2, and do some algebra from there, but that's...
The pit trap is that total average speed is not the average of speeds. If you're trained to pick a formula, you probably see "average" and pick (a+b)/2, and do some algebra from there, but that's the wrong formula.
Average speed is total distance over total time. On a 60mi trip, for the average speed to be 60mph, you have a time "budget" of one hour.
But if you traveled from A to B (30mi) at 30mph, you already spent that hour on the first leg of the journey. You have no more time left in the budget to make your average speed be 60mph.
E: looks like this comment is saying the same thing I did. This comment is a bit rude, but it's another intuitive way to see it. Pick more extreme numbers.
If you travel the first leg at 1mph, you already spent 30 hours an the trip. There is no speed you could travel on the second leg that would make the average speed be 60mph. At best, instant teleportation on the return would give an average speed for the entire trip of 2mph.
! Okay I see where I went wrong. (with embarrassment) Putting this up here for discussion on how people get things wrong. pretend the traveller is allowed to "overshoot" Aliceville. Let's say all...
! Okay I see where I went wrong. (with embarrassment) Putting this up here for discussion on how people get things wrong.
pretend the traveller is allowed to "overshoot" Aliceville. Let's say all distances are 30 miles between towns.
They drive A to B @ 30mph using the full first hour.
On the return, they drive from B, past A, through C and stopping at D. Distance from B to D is 90 miles total, so 90mph, using up a full second hour.
Average speed is total distance over total time.
So the average speed for this extended trip is (30 miles + 90 miles) / (2 hours) = 60 mph. This is where I landed.
Intuitively, I understood that if one goes faster on the second leg, the average mph goes up. With even longer distances traveled at a faster constant velocity, the average goes up even higher. Let's say traveller keeps going and going at 90mph:
(30 miles + 90 miles ) / 2 hours = 60mph
(30 miles + 90 miles + 90 miles ) / 3 hours = 70mph
(30 miles + 90 miles + 90 miles + 90 miles) / 4 hours = 75mph
But the reverse is true : if the trip is a short distance one, there's less pavement for the traveller to "make up for loss time". If traveller must stop at Canuckistan without going onwards to Delta:
(30 miles + 60 miles ) / 1.6666 hours = 54 mph
Therefore, where I went wrong is here: Even though intuitively, one could make up for slower mph by traveling faster at the next leg, this is only true at some distances. There's a minimum distance + time combination required to achieve certain avg speeds. I understood that the traveler could use more than 1 hour, we'll just divide it out, but I couldn't intuitively factor in the distance they must keep traveling at that velocity for the average to increase to the required rate.
thanks for being kind and patient and took the time with me :) exactly what kids in school need from their teachers
i think the problem is that you are trying to take an average of a continuous quantity, but you can't really reason about that without calculus (and then measure theory i don't understand yet...
Maybe it has to do with mph being a rate and adding and averaging rates is less intuitive?
i think the problem is that you are trying to take an average of a continuous quantity, but you can't really reason about that without calculus (and then measure theory i don't understand yet ;-;), so if you don't know calculus, you're sol. you can sort of handwave that away for problems that look specifically like this (where speed is a piecewise constant function), but the whole question falls apart if you allow speed to vary continuously at all. you have a symbol-pushing intuition that 'average' means a+b+c/3—of course that leads you astray, because that formula only means anything for a discrete problem
that makes me question whether it's even a good question. if we don't actually understand what a continuous average is, what are we to do? sibling says 'average speed is total distance over total time', but why is that and what does it mean? follows is my attempt at a better intuition (i don't know how good it is):
what do we want an average (mean) to be? in general: an expected value. suppose a bag has a bunch of lottery tickets in it, each worth a different amount of money; if you take out a lottery ticket, you get the corresponding amount of money, and replace the ticket. the average (lottery tickets are discrete, so we know how to do this) of the ticket values is also the amount we expect to make when we draw out one ticket. it won't be perfect, but it will be a good estimate. now suppose we draw a lot of lottery tickets; the more we take, the better an estimate it will give of how much money we've made
we'd like an average speed (whatever that means) to have a similar property. take any leg of the trip; it will cover some distance over some time, but suppose we only know one of those two. if we know only the distance of that leg, then the average speed should give us an estimate of the time it took us; and if we know only how long it took, then the average speed should give us an estimate of the distance. like with the lottery example, as we consider larger and larger legs (more time or distance), we should expect the estimate from the average speed to get better and better
but with the lottery example, we can just keep on taking more and more lottery tickets forever, and expect the estimate to keep getting better and better (it's perfect 'at infinity'). in this case, the trip has a natural starting and stopping point; there's no leg of the trip that takes longer than 'the whole trip' or goes farther than 'the entire distance of the trip'. so we should expect the average speed to become perfect not at infinity, but at the whole trip—given the average speed, the overall time should give us a perfect estimate of the overall distance, and vice versa. but that also means the average speed is determined exactly by the overall time and the overall distance
after that it's straightforward algebra (back to symbolpushing😔) to show that you'd have to make the trip back from B to A in no time at all to get your desired average, so it's impossible
After walking around a bit and having dinner, I've decided the question irks me and I want to come back to this thread to posit that this isn't a fair question at all. First, it's set up like the...
After walking around a bit and having dinner, I've decided the question irks me and I want to come back to this thread to posit that this isn't a fair question at all. First, it's set up like the "two trains leave the station..." problem, which in its most basic form I believe was just a "solve for X" kind of basic algebra situation.
This isn't that. This question then does two things that are meant to trick the reader. The first is that the only Arabic numerals used in the question are either 30 or 60. But there's another number that's expressly stated, but at least in English is obscured by being written out in a specific way. The other thing the question does is mostly talk about distance, miles, which is fortunately the sole unit of distance measurement, but then it also says mph. The tricky part about mph is that a lot of people think of it as a unit of measurement (it's the speed to match on the highway), but it's really two measurements and not one. Distance over time.
The number 1 is used throughout the question, but it's not notated in Arabic numerals, so that's a bit tricky and I think there's a phenomena where the human brain discounts the importance of non-numeral quantities or something. Combined with the "mph" part, and you never see the number one. You just see "per hour" all over. But it means 60/1 or 30/1.
This question would be much easier to mentally balance, and to parse mathematically, if it were written out like (30 miles/1 hour) and (60 miles/1 hour). In my head I want to express this question in the form of Stoichiometry and doing it that way it immediately becomes clear that there's a solve for X situation where there shouldn't be because the equation (info you have to work with) is imbalanced. You end up with a situation where you have something like (60 miles/1 hour) -> (X miles/0 hours remaining) and you have to divide by zero.
Of course not! It could be rephrased in any other way to make it more clear. A small diagram makes the issue obvious. I really like the word "pit trap" that chocobean used, because that's exactly...
this isn't a fair question at all.
Of course not! It could be rephrased in any other way to make it more clear. A small diagram makes the issue obvious. I really like the word "pit trap" that chocobean used, because that's exactly what it is. The problem sets up all the appearances of a simple solve-for-x problem, only for it to fall out from under you.
I'd say "problem" is not a good word for it at all. "Trick question" or "riddle" are more accurate, it's designed to deceive the reader. I personally think, if you approach it that way, it can be fun to try to find the trick. I can't imagine it feels fair if you're not expecting a trick, though.
I don't think you need calculus for this at all, and no handwaving necessary. You need calculus for continuous variations in speed, but for piecewise-constant speed (as the problem presents) you...
i think the problem is that you are trying to take an average of a continuous quantity, but you can't really reason about that without calculus. [...] you can sort of handwave that away for problems that look specifically like this (where speed is a piecewise constant function), but the whole question falls apart if you allow speed to vary continuously at all.
I don't think you need calculus for this at all, and no handwaving necessary. You need calculus for continuous variations in speed, but for piecewise-constant speed (as the problem presents) you can do perfectly fine with a position-over-time plot.
You want the average speed over the journey to be 60 mph, so draw that line as your "target" over the 60 mile trip. You're given that you travel from A to B at 30 miles, so draw that line too, at half the slope. Problem! You need a vertical line to get to the endpoint, which is not possible. Diagram.
sibling says 'average speed is total distance over total time', but why is that and what does it mean?
I think this really gets at the heart of the matter, and also the philosophy of why I say you don't need calculus at all.
Speed is all about motion, which necessarily involves measuring positions at multiple times. If you don't have multiple data points, it fundamentally doesn't make sense to talk about speed.
More specifically, if I have two data points:
I'm at position A at time 0h
I'm at position B at time 1h
Then my speed is a property of both measurements together. We define speed as the difference of positions divided by the difference of the times, it requires exactly two measurements. (Really that's velocity, but the distinction doesn't matter here.)
We can get more detail by taking more measurements along the journey but again, my speed is only really defined for pairs of position+time measurements.
As for averages, there are all sorts of averages. In all cases the goal is to summarize data. You have a big collection of data points, and you want to reduce it to a single representative value. The arithmetic mean is the fancy name for the canonical "average" that most people use most of the time. In most situations it does give a useful representative value; but it requires you to compare apples to apples. Everything has to be in the same units with the same weight, else you bias the result too far one way or the other.
So the usual meaning for average speed is to discard those intermediate measurements. We want to reduce the entire journey to exactly one speed, defined by exactly two measurements, summarizing the whole: the speed formula applied only to the start and end measurements, total distance / total time.
The great insight of calculus is that if you consider measurements that are very close together, things work out unreasonably well to introduce the paradoxical idea of instantaneous speed, and declare that it's whatever you calculate from measurements very close to that instant. But this is an invention by calculus, and since we are already given all data points we need by the problem, it is not necessary.
you have a symbol-pushing intuition that 'average' means a+b+c/3—of course that leads you astray, because that formula only means anything for a discrete problem
It's not about things being discrete, it's about things having the same weight. If we look at the diagram for 30mph depart and 90mph return, it's hopefully clear you can't just add the speeds together - the lines are different lengths!
So we can define a weighted average to try to correct this: multiply each value by its weight (length) to bring everything into common units. Now we're comparing apples to apples and can sum the weighted values. Then undo the adjustment by dividing again by the total weight (length). Note this is just a generalization of the arithmetic mean when both values have equal weight 1; written explicitly it's (1*a + 1*b) / (1 + 1) = (a + b) / 2.
So we have (1hr * 30mph + 20min * 90mph) / (1hr + 20min), and indeed that's 45mph.
But hold on - 1hr * 30mph has units of distance, that's 30mi. 20min * 90mph is 30mi. That's just the total distance! And 1hr + 20min - that's just the total time!
So this weighted average of speeds is just the usual average speed formula, total distance / total time. You can add as many measurements as you like, the inner terms always cancel and you end up with this formula.
And indeed if you consider instantaneous speeds and infinitesimal distances, the weighted average goes from a sum(speed * time) / (total time) to integral(instantaneous speed * dt) / (total time) which is exactly the usual formula.
I can't shake this feeling that we're just teaching maths wrong? I keep coming back to this Numberphile video which compares maths to art class, that we don't try to inspire maths in the same way.
I don't necessarily buy the argument of maths having a higher barrier to entry compared to great art or literature either, done right I think it's possible to get a curious person interested in mathematics without scary formulas and concepts.
we are obviously teaching everything wrong
i don't know what is in that video, but classic wrt maths is mathematician's lament https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician's_Lament.pdf
I've never understood why all the cool things that you can do with math aren't part of how it's taught. All the math problems when I was a kid were about divying up ridiculous amounts of produce or measuring the heights of random buildings that no kid in that class could possibly care about. I know the mathematical concepts needed to build things or understand a lot of the actually fun things are too advanced for most kids, but it's not impossible to explain how those things are going to be used as building blocks. Heck, make the kids read the mathiest days from XKCD Explained and use that as a springboard.
I'm not saying it's easy, but at least when I was a kid there seemed to be extremely little realistic effort to make math interesting or enjoyable to kids. Not zero effort, there were those word problems, but none of them felt like any thought had gone into what kids might want to use math for.
I only have experience teaching grade 8 algebra with one kid, so survivorship bias and extremely unreliable sample size warning.
The kid gets frustrated with tiresome busy work very quickly, as would we all. So the curriculum has to rotate out types of problems and mix new with varying kinds of old.
Humour helps: make the word problems or situations extremely silly, such as a dog wants to sniff X number of butts or I want to stack Y number of skittles on the cat. Make them completely absurd. Your point on XKCD and explained exactly. So many of Randall's videos of what ifs solve absurd problems using a lot of maths for just plain silly fun.
Usefulness is also underrated. Baking with imperial units of measurement, scaling recipes up or down, min maximizing the quantity of sugar cereal you can buy by calculating the mass per dollar, EV training your pokemon, watching savings account interest NOT grow perceptibly against inflation vs equity...
You can't build a skyscraper without a foundation, so there's going to be boring bits, but get a good balance of old/new/humour/relevance in there and it's very doable. The trouble with public education of maths has always been and will continue to be trying to find that balance, but now with 30 kids all at different levels of foundation and feelings and interests.
Parents have to retake the reins on this one I think. School has been proven to be terrible at teaching maths. You gotta do it yourself
Parents who mostly learned math in those same schools except they haven't brushed up on any of it in decades? This is a ridiculous ask. I'm the sort of person who watches Numberphile and has an M.Sc, and I'd still be pretty bad at teaching a kid math. And most parents have less math literacy and less time than I do.
I'll tack onto this: I took a class in college about teaching math to elementary school students. The instructor was a very passionate man about education and math, one of the best teachers I've ever had. Half the reason I took that class is because I had him for my first-semester math class, and he made it so much easier and more enjoyable than any other math class I've ever taken.
He told us an anecdote about having to sit in on an exam for another teacher once, and he decided to glance at the exam while the students worked on it. Even though he had taught that subject in the past, he couldn't remember how to solve any of the problems because it had been a few years since he'd taught it.
If a deficated and passionate math teacher can forget how to solve certain problems after a few years, I wouldn't expect parents to remember after 20+ years.
I'm at a loss myself, in all honesty, on how to do it myself, if by "it" we mean how to teach math properly and how to become a numberphile and see beauty in it.
All I can aim for is a healthy happy child who can push through the ugly slog, coming out alive and competent enough to get their degree and never look at math again.
I'm definitely not claiming that I can be the sort of mathematician that The Lament wants, someone to inspire love and wonder and appreciation of math. All I can do is monitor individual attention, make daily adjustments, try to be firm but flexible, encouraging without coddling, and then requiring that the kid put in their hours doing the boring, soul crushing, meaningless and ugly math that society is requiring of them.
I hate it, I wish it wasnt this way, but we will be able to push through and so have millions of parents. Math has to stop being presented as a sort of elective because this sick and disgusting society isn't treating it this way.
My parents never graduated elementary school and have never done algebra, but they paid for tutoring to make sure we can do it. There's a weird American culture that drop off kids at school and pretend that's all parents need to educate: this attitude has to change.
No one has to love math: this is a tragedy. But also parents need to stop expecting the school will do it either. Mr Incredible spent a lot of time sitting with his kid doing "new math" -- it isn't the curriculum, it's spending the hours doing the ugly boring irrelevant math together. Seeing your parent struggle, complain together, but persisting -- that, I believe, is more realistic and more helpful than asking more of our teachers who are already tasked beyond the impossible
Having the time to tutor your kid yourself or the money to hire someone else to do it is a privilege, though, and as a result relying on parents to teach their kids math only reinforces existing inequalities. The entire point of free public schools is that students can be educated to the same standards (at least ideally) even when their parents have vastly different resources. The way public schooling is funded in the US already undermines this goal, since schools are mostly funded based on the wealth in their districts, but relying on parents rather than teachers only exacerbates that same issue. Deciding to teach your own kid math at home is perfectly reasonable as a choice to optimize things for them specifically as an individual, but making that the general solution upon which we base decisions and public policy is bad for society.
You insist that asking teachers to do more when they're already asked to do the impossible is not viable, but the reason that it's impossible is because there are far too few teachers and they're paid a pittance. Those are both problems that could be solved through actually properly funding schools and paying teachers a wage that even kind of approaches their societal value. You can argue that's unrealistic in our current political climate, but it's no more unrealistic than expecting kids to get math tutoring outside of school when their parents are working multiple jobs to make ends meet while living paycheck to paycheck. Unless you fundamentally don't care about kids having grossly different opportunities and education based on their parents' income, it's actually a lot more feasible to pay teachers better than it is to ensure all parents are capable of teaching their children math at home.
100% in agreement with you. I wish funding education was the priority that it should, and teaching wages was the focus and lion's share of these resources, not buying more laptops or building more gyms or whatever "hardware" "feels" more valuable than just - plain - wages - for - good - people.
I might sound like I'm contradicting myself : speaking as a citizen of how we should improve a system often comes in conflict with how I should parent my one child here an now who is only a child for x number of years. What I think society should do is often diametrically opposed to advice I would give to actual on the ground parents who don't have years/decades/votes to change the system -- I think it's evil that we're having to choose between voting with our kids' limited time in public vs private / tutor, and how these things then get funded.
It's like asking for people to stop driving and take transit, because if no one takes transit it won't get funded, but no one can take transit right now because it's terrible. I think these are intentionally manufactured problems and politically I will always vote / volunteer to change them, even as I live as part of the problem. =..= very depressing...
Just to affirm what you’re saying, having been a former Kaplan test prep instructor, my takeaway was that the main value add of those courses has less to do with the curriculum and more the amount of practice they force students to do. The practice tests and problems build up the mental and physical stamina to be able to sit still for multiple hours to grind through the exam. Most people who did fine in school ought to know the majority of the content, it’s just that they tire out. Especially on the reading comprehension and problem solving elements that require focus.
We expect everything we learn to be fun and inspiring, but sometimes you just have to clear a hurdle by getting a lot of reps in through rote memorization. We don’t doubt this when we talk about fun things. If you want to get good at basketball you practice free throws to develop a sense for how to throw the ball. If you want to get good at piano you have to practice until you can reach the notes instinctively. If you want to get good at video games you play enough to where you know the layout of the map and can have fast enough twitch reflexes. And if you want to be good at math it sure helps to just be able to know, reflexively, that 9x9 is 81 without needing to stop and check or calculate. Getting the mental calculation smooth enough to where it doesn’t introduce friction into your reasoning process is what actually lets you unlock more of your brain’s capacity to do the more abstract reasoning you need for algebra and calculus.
Part of the problem is that most parents are also terrible at math. And I think a lot of people have a sort of mental block with it. Being Indian I notice a similar effect with my name. I usually just give my initials at coffee places to avoid mispronunciation and more than half the time people ask me “How do I spell it?” The brains are on autopilot to where they see an Indian person and just assume whatever comes out of my mouth when they ask my name is going to be unpronounceable so they try to skip ahead and don’t processing that I just told them two letters. A lot of people do this with any kind of arithmetic where they just see numbers of any kind (and God forbid those numbers be mixed with symbols) and their brains check out.
Bolded my change. While what you said is correct, I have met far too many elementary teachers (or aspiring teachers in my college years) with math anxiety. Math is not some insurmountable tricky magic; it's just math.
I'd also add that much of what makes students hate math is arithmetic. They get bored and frustrated, so they check out, thus causing gaps in their knowledge that make later topics actually insurmountably challenging. Math is unique in how rigid its prerequisite structure is—there is no need to start the study of history with the speciation of H. sapiens.
yes! Exactly! Both that (1) there's a "muscle" to the very fundamentals, and (2) cultural mental block
There's a ditch on either side of the road: kids will kill themselves if pushed too hard, but kids will also default to not ever automatically knowing 9x9=81 if they didn't spend enough time for their brain to register it into longer term memory.
I greatly enjoyed Mathematician's Lament, but at the same time I reflect back on my years spent in Canadian high school art and band classes, and how much of a waste they were. I was never taught colour theory, art history, circle of fifths, transposition or any of the boring bits. Everything was just a circle-praise of "wow, I love this, A+" and be free and here's more empty canvas! I did have fun, I appreciate some types of art and music, but I'm absolutely ruined for dedication and concentration and effort towards making good music and good art: "everything is music everything is art hurray!" Thank goodness I didn't need art and music to obtain a level of life where I, now as an adult, can choose to do more music and art out of pocket. So, credit: I came out not hating art/music, I love it.
Could the same thing happen? Do citizens who have a grand and lovely time with math in high school go on to enjoy math and participating in the thinking aspects beyond passively watching a video mathematician enthusiastically talk about math? I highly doubt it. It's okay to push a little bit, early and often and kindly.
Question: what were your Kaplan kids like? What about the kids whose parents didn't send them to tutors / who made such a fuss they withdrew -- could you teach them the same way or how do we teach them?
I was teaching the MCAT and I was, like, 2 years older than my students so it’s not the most typical teacher/student relationship. I also only had one cohort of students. It’s a medical school admissions test so they tended towards being highly motivated gunners who had different levels of heart in it. Everyone was motivated to do well on the test for that cohort so the discipline was mostly there, but there was a decent number whose heart just wasn’t in it. They were already burned out before even being done with their bachelors degree so you can tell that even if they’re smart enough to score well they’re not gonna make it all the way through the slog that is medical school.
Hmmm interesting sample...... highly motivated young adults who chose this program with the math requirements is a very different circus than 8th graders who gain social benefits from complaining about math and rebelling against being told to do things in school.
I don't know that parents can be expected to have the resources to do that. Teaching these concepts well, in an interesting way, takes a much more multifaceted and nuanced skillset than teaching them in a boring, frustrating way, and that's assuming that they have grasp of the basic math to teach them these concepts at all.
One of the benefits of a school system should be the variety of disciplines that can be pulled from to work together in the creation of curricula like this. Obviously teachers aren't given the resources required to pull this off, or the respect they deserve when they manage to do it, I am absolutely not blaming teachers en masse for this. Parents don't have all of those resources to pull from, and they have other jobs on top of it.
I know it's not realistic to expect this from our public school systems, but it's also not unreasonable.
I completely agree that not all parents have, frankly, this luxury to either do it themselves or hire a tutor who could. Then we circle back to an equity issue, and if we then multiply it by the horror of requiring calculus / discreet maths for a high paying job, essentially we have an all in one package to perpetuate cycle of poverty, blaming the citizens for choosing to not try hard enough at math.
I did grow up with the East Asian system of pushing super hard: at least two hours a day [edit: 6 days a week plus at least 1 hr home work a day for just math] for kids who get it, mandatory afterschool tutoring for kids who don't, and "optional" afterschool tutoring for kids whose parents will require post sec education of them. The down sides are laid out in @Moonchild 's mathematicians lament. But that super long (excellent) essay is forgetting the "up sides" (shudder) of pushing far harder than Americans systems are prepared to do.
Strong air quotes around up sides. Very strong. Kids who don't kill themselves, literally, will be able to do algebra and post sec math and get good jobs. All those college entry boards know this: they see far more well pushed kids than kids who "are interested" in math naturally.
Fact is. Society is sick, and has thrown up these hurdles between kids and a financially secure future. We can either change society or we can bend our kids to fit into it. Some kids will break instead of bend, but most kids will bend with enough pressure. What this one states is doing is wasted effort; it's pushing without pushing enough, making everyone miserable without fitting kids into the mandated slots. It's like bad braces or starving followed by binge.
The real problem is society.
That sounds horrible. I’m glad I was able to enjoy math within the system. I want to give credit to the handful of math teachers at my public high school that were excellent. My statistics teacher loved the subject and teaching so much she’d occasionally jump up and down with excitement during a lesson. My calculus teacher was one of the kids’ favorite in the whole school. He managed to make it fun while caring about each student and explaining things well.
In HK, I got the ruler a lot. On a good day it's smacks to the palm, not the back. Social shame was a popular one. All kinds of schoolyard and domestic abuse that hopefully don't happen anymore.
In a Canadian Highschool - I've had a few really excellent math teachers as well. One of them has a (small) roster of repeated dad jokes and sometimes tries to freehand draw the perfect 1m+ diameter circle on the white board -- the lameness of jokes works GREAT on us to collectively groan in solidarity, and we just LOVE to nit pick where a circle might not be perfect. Another was just a great dad type figure, so we respect him when he says "alright, lets' get back to X". Another one is a jolly woman called us her little chickadees and has an infectious laugh. Attitude and excitement are infectious, indeed. Kindness helps. Patience definitely.
But even those wizards wouldn't have been able to do much with a class full of kids who are just marking their time, singing and spacing out and talking, like the 8th graders in the article. If the prevailing culture climate is "math is hard/dumb/useless - pass all the ones you can, go to Track A if not, and beg for C+ until they're no longer compulsory", kids won't put in the mental work to concentrate enough to actually benefit from their kind wizardry.
The braces thing is a really good analogy. I think a lot of people just want things to suck for suckiness’ sake, like it will forge them into being stronger. But the stress is a cost you pay to be able to get through, it’s not the thing of value in itself.
On the point of usefulness, something that I think gets ignored by curriculums far too often is that with how some peoples’ minds work (kids and adults alike), for subjects that the individual doesn’t find inherently interesting, usefulness isn’t just a nice-to-have, it’s nearly a requirement.
I’m like this. The difference between how deeply and quickly I learn something that I need to know in order to accomplish a task vs. something that feels arbitrarily foisted upon me by a course with no immediate purpose other than to pass an exam is staggering. As an adult I have the patience and discipline to slog through the latter (though I’ll still hate it), but in middle/high school and uni it was a very different story. Naturally, this made all maths from algebra up a struggle for me, despite having a good enough grasp on logic to wind up working as a programmer.
Schools all the way from elementary up through high school don’t do anywhere near enough hands-on to make practical use of learnings. Uni is a bit better but still has plenty of room for improvement there.
agreed.
In one episode of Clarkson's Farm, Caleb admits to having been absolutely terrible at math. But when he approaches a problem in the form of farming, a vocation he loves, he was able to work out nearly instantly how much money could be made if x grain was purchased at y price and planted in z acres with harvest rate of a% sold at b dollars per c quantity. If school had been like this for Caleb, he would have absolutely been an A maths student.
In my experience as a student anyone with a desire to teach mathematics has also a strong desire to avoid anything soft and squishy like emotion and inspiration. It might be easier to train art majors to teach maths than to persuade a "real" math teacher to care about inspiring anyone who is not in love with math already.
I've definitely been fortunate that many of my maths/science teachers weren't jaded and boring, but nonetheless I think mathematicians have a deep love of "squishy" things like beauty and comedy.
They talk frequently about equations or proofs being beautiful (e.g. Euler's identity), and the writers for Simpsons/Futurama are accomplished mathematicians.
I would propose what you experienced was people who loved math and were required to teach.
Students regularly had this idea that teaching was what I spent my time doing when in reality it was a handful of hours a week. I liked teaching but many of my peers did not.
That is what I thought of teachers coming from engineering or physics, but some of the worst math teachers I had were actual mathematics majors with licenciature (meaning they:already majored as educators) or "magistério" (a course you can take in Brazil to become a teacher).
They where all so bad. At least Humanities and Arts teachers showed passion sometimes. Math teachers were dead inside lol
Math is a subject that requires deep work to master, lending itself to individual and small group formats. Reading, re-reading, applying in practice, watching examples, and formulating questions is necessary as you get into calculus.
While I believe that every person is capable of developing these habits and applying them to math, not everyone is ready at the same time. Kids in particular develop in different ways at different ages. Until someone is developed enough to have the ability to work individually to some extent, you really can't force understanding on them. There's no amount of rote demonstration that will convey understanding of epsilon-delta proofs, for example, and those are week 2 or 3 of a calculus course (if you are teaching for material mastery). So if your goal is more people who take calculus successfully, you need a mechanism to assess when they are ready to start learning the concepts that are beyond the "math facts" variety.
While having an engaging curriculum is helpful, there is no set of exercises that will get a student to a point of proficiency without them having some ability to study and learn. So we should focus on making sure we teach kids how to learn, and not lock them out of opportunities just because they developed those skills a little later than their peers. More emphasis on competency based placement in K-12 courses rather than age-based might help, but it is a difficult problem to fix within the confines of our schooling system and the lack of resources many students have at home.
This is one of the reasons I like the Montessori method for teaching. Classes combine students from three years together. At my daughter's current school, this means 3yo-kindergarten, 1st-3rd grade, and 4th-6th grade.
This makes room for the "sensitive periods" so that when interest sparks on something, the student can deep dive into it. But an equally important piece is that the younger kids see what the older kids are doing, so they have a sense of where they are headed. As they move up, they become the leaders in the classroom and often teach the younger kids, which provides reinforcement.
The work is also integrated so that math becomes part of other subjects, which allows them to apply what they have learned in practical ways.
My daughter is only in the fourth grade, and I only did Montessori through second grade.
So I don't know how well the methods translate into high school subjects like algebra and calculus or the more categorically focused sciences (biology, chemistry, physics). There's no Montessori high school available locally anyway.
My hope is that she will have enough foundational knowledge, confidence in her ability, and willingness to ask for help that she can excel in a more conventional environment for middle (generally 7-8th grade locally) and high school (9th-12th grade).
One thing that's difficult applying these lessons to conventional schools (speaking from a US perspective) is that the structure is radically different, so a single teacher's ability to implement these principles is limited, both because they have little control over their curriculum and because they need to prepare students for a specific next class. It also requires different training and a different mindset from the teachers, though from my observation, it's more fulfilling for the teachers overall.
The specific topic of epsilon-delta proofs may be one of the few topics that aren't an actual blockage for students who don't get it in the allocated curriculum time. They're necessary so students don't take the theorems and derivative formulas by faith, but not understanding them doesn't preclude understanding the application of limits and integrals.
Contrast with most of the rest of math, where there is a strong sequence that must be followed to build understanding.
Are epsilon-delta proofs just a fancy way of saying "learning how limits work"? I took calc 1 and did very well, but I have no idea what an epsilon-delta proof is. But at the stage of calc 1 they're referring to, we were learning how to work with limits, which I recall making the subsequent move to derivatives seem very intuitive, so I'm wondering if the same material was covered without using that terminology.
You are correct. Epsilon Delta proofs are the mechanics of limits. They are often touched on at the start of calculus one, and only re-covered in the first semester of real analysis which is largely limited to math majors and some smattering of STEM folks who go hard on the theory side of things. They are important to cover conceptually, but unwieldy to apply.
Edit: they also notoriously give students fits as the first and sometimes only exposure to thinking in terms of proofs.
I don't recall whether I actually learned any proofs specifically when I learned limits in calc one. I definitely learned the intuitive concept of limits, but it was over 10 years ago so I may just not remember the proofs side that clearly. We definitely didn't do epsilon-delta proofs in the refresher course I took in the first semester of my M.Sc, but that course was trying to combine calc 1, basic stats, and linear algebra all into one semester so we could handle machine learning in the spring, so everything was at least a little rushed.
It's a tough curriculum choice to make. Many college level courses for stem degrees will cover them in a lecture, maybe have some problems on the homework, but not test them. Other programs might skip them entirely, especially if they don't have separate stem and non-stem calculus courses.
Personally I like teaching them because if nothing else it exposed students to a type of thinking they might not otherwise see. Putting them on a homework, but not testing seems like a decent compromise.
The downside is that math programs tend to push more and more of the "real" math to high level courses, which can create a fourth year weed out of real analysis, which is a harsh situation for math majors to have.
Edit: I don't think they are covered in AP calculus, but I could be misremembering. Most of the students I spoke to didn't see them until college even if they did AP calculus.
I took calculus in a post-secondary enrollment program (where I took classes at a local college as a highschool student) rather than taking AP courses, so it was dependent on the whims of my specific math professor essentially (and I learned a lot from that class and it was the first math course I'd ever taken I didn't hate, so kudos to her for that.)
It's always great to hear about good teachers. I used to dislike math until I had a great calculus professor, and that led me to getting degrees in math. It's amazing what a good presentation of the material can do.
One of my regrets is that I let my dad convince me to take accounting the subsequent semester instead of trying out number theory like that prof suggested. I think that might've led to me doing more real math in college if I'd found it interesting, whereas accounting was incredibly easy but equally boring. But I ended up doing linguistics and focusing on formal semantics, which involves a bunch of set theory and lambda calculus, so I at least got something math-related in there eventually.
I picked those proofs because they defy rote memorization, like most proofs. But I would say that most of the topics in calculus can't really be absorbed or mastered through rote memorization. Even things as simple as the application of the chain rule takes an understanding of the process in order to apply to arbitrary problems. I suppose you could memorize algorithmic steps for much of calculus, but suspect you would get destroyed on any real test of the material that asks you to apply calculus to a situation. E.g., using it on the context of physics or other sciences.
Developing intuition for rates of change, rates of rates of change, inflection points, families of solutions, etc. are important to any real mastery of the material. And those are the things that can't really be rushed in the K-12 space. Kids need to have developed the healthy mental habits to foster learning in order to really learn these topics.
My teachers called it number sense. In a less advanced situation, it’s the gut instincts that tell you when you miskeyed the arithmetic in your calculator.
I never heard it called that, but I had some teachers teach me exactly this. First, does the answer make sense in the context of the question? If you calculate the moles of a substance and get a negative number, you did something wrong. Then, you can do the problem in your head considering only the magnitude. If your mental math says you should be around 8 orders of magnitude and your answer is 3689, you definitely did something wrong. I am sure there are other tricks, but those two saved me a lot of wrong answers. Also the magnitude only calculations are surprisingly useful in daily life.
I learned integration by parts this way in high school, and indeed I was destroyed by it on tests. I (and most of my peers from what I recall) did our best to memorize the algorithm and try to recognize when/where to apply it, but I always found that recognition impossible. Differential equations in undergrad was awful for similar reasons, being handed some list of divine algorithms and told to, somehow, recognize which one to use in a given situation. Again, I was destroyed. My only D in undergrad, but hey, you can graduate with a few D's and life moves on.
It is not until graduate school that a professor makes an offhanded comment that we'll need integration by parts, and that's just product rule. Earth-shattering revelation; integration by parts feels easy now. I really wish someone had said so in high school, but instead the focus was on this rote memorization of algorithms. Or, if someone did say so, I wish I had paid attention. I often wonder if I was told things like that, but I just lacked experience or intuition at the time to recognize the value of it.
It's amazing what the right presentation of a method or idea can do for understanding. Differential equations is especially notorious for crushing students and forcing desperate memorization, though I blame some of that on overly full curriculums.
Cheers!
I have another thesis: It's not age, it's cumulative hours spent on maths prior to algebra.
What this state is doing seems to be offering algebra by 8th grade. This created an awkward gap between too early and way too lax. You don't just offer maths, you have to make it compulsory early and often so that being able to do basic pre algebra maths is an achievable requirement. If kids are already struggling with understanding scientific notation or don't have their multiplication and division effortlessly, that's too early. But by the time they're 13+ they're so far behind that doing the calc route becomes nearly impossible to catch up to kids who put in the cumulative hours in their primary years.
Elementary school maths is laughable. What are they doing. I know they want to discourage rout learning and memorizing, and they want to emphasize higher reasoning etc. but at some point it's like trying to build a car using hand fashioned tools made of clay: there's so little time to develop the automated portions that the basic tools required to quickly and painlessly do algebra geometry calculus just aren't there.
It takes time. Time the primary years should have accounted for. The tragedy here isnt pushing algebra, it's not enough pushing prior to algebra.
You can't get there from couple hours of easy loosely goosey fun maths a week. You gotta have smaller class and individual tutoring and sheer time spent doing the basics. Judging by how few American adults choose math, the few who do move onto tech jobs -- who would want to teach math when it barely pays enough to live / student loans. It's a complex problem decades in the making.
If that's the biggest barrier I guess the entire education system is in danger. And always has been for as long as we had structured education. But I digress.
It's an interesting idea, but mathematics is an area where fundamentals are everything. if you're 1st-7th graded mathematics aren't rock solid, you're going only see an amplification of struggles for algebra. I'm sure there's a myriad of factors that can improve in this regard, but it ultimately comes down to each individual everyone has different proficiencies, and rushing them is the last way to help here.
For reference, I did do an accelerated track and I'll admit that I struggled a bit in the hybrid Algebra II/Calculus A course. You know what helped the most in Calc B/C? The teacher took 2 weeks to reinforce our trigonometry . I can't really explain what "clicked" that 2nd time, but seeing trig again after that exposure to how growth and limits work kinda just did something. I suppose the first time through I was relying a lot on rote memorization (and tbf, a good part of trig is just accepting seemingly random identities). The 2nd time I could understand more of the why's and how's behind how the curves are modeled, and received more pratical applications of what they are used for besides triangles.
I'm pretty sure I was taught these concepts in eighth grade, back in the day (in Portugal). It was one of the only two classes I ever had a negative grade in (the other being 5th grade English), and the teacher was pretty decent. Eventually he gave me some personal tutoring and I got past that hurdle, then it was smooth sailing all the way to the end of high school (calculus in high school here ended with basic limits and derivatives).
I'd say these lessons can be tough to tackle for the first time in your life but they're so important and fundamental that I'm absolutely behind their early introduction. IIRC most of my classmates got through it, and even an idiot like me! No phones or laptops at the time, though (this definitely stands out in the article).
Tangential:
Saw this math problem on Reddit a few days ago:
the answer is .... ? Right?
90mph on the return trip right?Most everyone says it's impossible and you'd have to travel at the speed of light and still not increase the velocity to an average of 60 mph.
If folks are wrong, how are this many people wrong? If they're right can someone explain this to me? One possibility is that people (potentially myself included? ) got trained by school math to pick a formula, then plunk numbers in and viola. There must be some common pit trap here where people are trained to automatically picking a bad formula to begin with with? Maybe it has to do with mph being a rate and adding and averaging rates is less intuitive?
The pit trap is that total average speed is not the average of speeds. If you're trained to pick a formula, you probably see "average" and pick
(a+b)/2, and do some algebra from there, but that's the wrong formula.Average speed is total distance over total time. On a 60mi trip, for the average speed to be 60mph, you have a time "budget" of one hour.
But if you traveled from A to B (30mi) at 30mph, you already spent that hour on the first leg of the journey. You have no more time left in the budget to make your average speed be 60mph.
E: looks like this comment is saying the same thing I did. This comment is a bit rude, but it's another intuitive way to see it. Pick more extreme numbers.
If you travel the first leg at 1mph, you already spent 30 hours an the trip. There is no speed you could travel on the second leg that would make the average speed be 60mph. At best, instant teleportation on the return would give an average speed for the entire trip of 2mph.
! Okay I see where I went wrong. (with embarrassment) Putting this up here for discussion on how people get things wrong.
pretend the traveller is allowed to "overshoot" Aliceville. Let's say all distances are 30 miles between towns.
They drive A to B @ 30mph using the full first hour.
On the return, they drive from B, past A, through C and stopping at D. Distance from B to D is 90 miles total, so 90mph, using up a full second hour.
So the average speed for this extended trip is (30 miles + 90 miles) / (2 hours) = 60 mph. This is where I landed.
Intuitively, I understood that if one goes faster on the second leg, the average mph goes up. With even longer distances traveled at a faster constant velocity, the average goes up even higher. Let's say traveller keeps going and going at 90mph:
But the reverse is true : if the trip is a short distance one, there's less pavement for the traveller to "make up for loss time". If traveller must stop at Canuckistan without going onwards to Delta:
Therefore, where I went wrong is here: Even though intuitively, one could make up for slower mph by traveling faster at the next leg, this is only true at some distances. There's a minimum distance + time combination required to achieve certain avg speeds. I understood that the traveler could use more than 1 hour, we'll just divide it out, but I couldn't intuitively factor in the distance they must keep traveling at that velocity for the average to increase to the required rate.
thanks for being kind and patient and took the time with me :) exactly what kids in school need from their teachers
i think the problem is that you are trying to take an average of a continuous quantity, but you can't really reason about that without calculus (and then measure theory i don't understand yet ;-;), so if you don't know calculus, you're sol. you can sort of handwave that away for problems that look specifically like this (where speed is a piecewise constant function), but the whole question falls apart if you allow speed to vary continuously at all. you have a symbol-pushing intuition that 'average' means a+b+c/3—of course that leads you astray, because that formula only means anything for a discrete problem
that makes me question whether it's even a good question. if we don't actually understand what a continuous average is, what are we to do? sibling says 'average speed is total distance over total time', but why is that and what does it mean? follows is my attempt at a better intuition (i don't know how good it is):
what do we want an average (mean) to be? in general: an expected value. suppose a bag has a bunch of lottery tickets in it, each worth a different amount of money; if you take out a lottery ticket, you get the corresponding amount of money, and replace the ticket. the average (lottery tickets are discrete, so we know how to do this) of the ticket values is also the amount we expect to make when we draw out one ticket. it won't be perfect, but it will be a good estimate. now suppose we draw a lot of lottery tickets; the more we take, the better an estimate it will give of how much money we've made
we'd like an average speed (whatever that means) to have a similar property. take any leg of the trip; it will cover some distance over some time, but suppose we only know one of those two. if we know only the distance of that leg, then the average speed should give us an estimate of the time it took us; and if we know only how long it took, then the average speed should give us an estimate of the distance. like with the lottery example, as we consider larger and larger legs (more time or distance), we should expect the estimate from the average speed to get better and better
but with the lottery example, we can just keep on taking more and more lottery tickets forever, and expect the estimate to keep getting better and better (it's perfect 'at infinity'). in this case, the trip has a natural starting and stopping point; there's no leg of the trip that takes longer than 'the whole trip' or goes farther than 'the entire distance of the trip'. so we should expect the average speed to become perfect not at infinity, but at the whole trip—given the average speed, the overall time should give us a perfect estimate of the overall distance, and vice versa. but that also means the average speed is determined exactly by the overall time and the overall distance
after that it's straightforward algebra (back to symbolpushing😔) to show that you'd have to make the trip back from B to A in no time at all to get your desired average, so it's impossible
After walking around a bit and having dinner, I've decided the question irks me and I want to come back to this thread to posit that this isn't a fair question at all. First, it's set up like the "two trains leave the station..." problem, which in its most basic form I believe was just a "solve for X" kind of basic algebra situation.
This isn't that. This question then does two things that are meant to trick the reader. The first is that the only Arabic numerals used in the question are either 30 or 60. But there's another number that's expressly stated, but at least in English is obscured by being written out in a specific way. The other thing the question does is mostly talk about distance, miles, which is fortunately the sole unit of distance measurement, but then it also says mph. The tricky part about mph is that a lot of people think of it as a unit of measurement (it's the speed to match on the highway), but it's really two measurements and not one. Distance over time.
The number 1 is used throughout the question, but it's not notated in Arabic numerals, so that's a bit tricky and I think there's a phenomena where the human brain discounts the importance of non-numeral quantities or something. Combined with the "mph" part, and you never see the number one. You just see "per hour" all over. But it means 60/1 or 30/1.
This question would be much easier to mentally balance, and to parse mathematically, if it were written out like (30 miles/1 hour) and (60 miles/1 hour). In my head I want to express this question in the form of Stoichiometry and doing it that way it immediately becomes clear that there's a solve for X situation where there shouldn't be because the equation (info you have to work with) is imbalanced. You end up with a situation where you have something like (60 miles/1 hour) -> (X miles/0 hours remaining) and you have to divide by zero.
Of course not! It could be rephrased in any other way to make it more clear. A small diagram makes the issue obvious. I really like the word "pit trap" that chocobean used, because that's exactly what it is. The problem sets up all the appearances of a simple solve-for-x problem, only for it to fall out from under you.
I'd say "problem" is not a good word for it at all. "Trick question" or "riddle" are more accurate, it's designed to deceive the reader. I personally think, if you approach it that way, it can be fun to try to find the trick. I can't imagine it feels fair if you're not expecting a trick, though.
I don't think you need calculus for this at all, and no handwaving necessary. You need calculus for continuous variations in speed, but for piecewise-constant speed (as the problem presents) you can do perfectly fine with a position-over-time plot.
You want the average speed over the journey to be 60 mph, so draw that line as your "target" over the 60 mile trip. You're given that you travel from A to B at 30 miles, so draw that line too, at half the slope. Problem! You need a vertical line to get to the endpoint, which is not possible. Diagram.
I think this really gets at the heart of the matter, and also the philosophy of why I say you don't need calculus at all.
Speed is all about motion, which necessarily involves measuring positions at multiple times. If you don't have multiple data points, it fundamentally doesn't make sense to talk about speed.
More specifically, if I have two data points:
Then my speed is a property of both measurements together. We define speed as the difference of positions divided by the difference of the times, it requires exactly two measurements. (Really that's velocity, but the distinction doesn't matter here.)
We can get more detail by taking more measurements along the journey but again, my speed is only really defined for pairs of position+time measurements.
As for averages, there are all sorts of averages. In all cases the goal is to summarize data. You have a big collection of data points, and you want to reduce it to a single representative value. The arithmetic mean is the fancy name for the canonical "average" that most people use most of the time. In most situations it does give a useful representative value; but it requires you to compare apples to apples. Everything has to be in the same units with the same weight, else you bias the result too far one way or the other.
So the usual meaning for average speed is to discard those intermediate measurements. We want to reduce the entire journey to exactly one speed, defined by exactly two measurements, summarizing the whole: the speed formula applied only to the start and end measurements, total distance / total time.
The great insight of calculus is that if you consider measurements that are very close together, things work out unreasonably well to introduce the paradoxical idea of instantaneous speed, and declare that it's whatever you calculate from measurements very close to that instant. But this is an invention by calculus, and since we are already given all data points we need by the problem, it is not necessary.
It's not about things being discrete, it's about things having the same weight. If we look at the diagram for 30mph depart and 90mph return, it's hopefully clear you can't just add the speeds together - the lines are different lengths!
So we can define a weighted average to try to correct this: multiply each value by its weight (length) to bring everything into common units. Now we're comparing apples to apples and can sum the weighted values. Then undo the adjustment by dividing again by the total weight (length). Note this is just a generalization of the arithmetic mean when both values have equal weight
1; written explicitly it's(1*a + 1*b) / (1 + 1) = (a + b) / 2.So we have
(1hr * 30mph + 20min * 90mph) / (1hr + 20min), and indeed that's 45mph.But hold on -
1hr * 30mphhas units of distance, that's30mi.20min * 90mphis30mi. That's just the total distance! And1hr + 20min- that's just the total time!So this weighted average of speeds is just the usual average speed formula,
total distance / total time. You can add as many measurements as you like, the inner terms always cancel and you end up with this formula.And indeed if you consider instantaneous speeds and infinitesimal distances, the weighted average goes from a
sum(speed * time) / (total time)tointegral(instantaneous speed * dt) / (total time)which is exactly the usual formula.