9 ms·

In Math Cram Sessions, Solving for Why

172 points by danso 8y ago

somethingroma 8y ago
"This was the key. If you knew hhhhwwhy you didn’t have to memorize equations, or solve equations in the exact same way they did in the book..." I feel like this is a valuable lesson a lot of people could benefit from.
- tesin 8y ago
  I agree as a generalization, but that does presume that the syllabus itself doesn't need rote interpretation. “No, we haven’t learned about some of the vectors yet, so for the sake of the problem, we just take this one out,” Many many students self-combust when faced with garbage worded questions because they never learnt how to deconstruct them in class. When questions come with spelling errors and ambiguous variables it's tough.
- Koshkin 8y ago
  Knowing "why" is exteremely helpful in many cases. But let's not forget that, on the other hand, memorization plays a huge part in learning, and not only in mathematics. Learning "why" multiplication works will not help you to retain the multiplication table in your head. (Also, in many cases the answer to a question "why" you might get will be plain wrong - sometimes because the correct answer is too complicated, sometimes for other reasons.)
  - somethingroma 8y ago
    Memorizing multiplication tables is over rated. Seeing connections between those early multiples up to 12 is more interesting. Can be helpful I suppose in factorization. Maybe I just don't trust my memory (as in how am I sure that 7*8 is 56) and why I despise rote memorization and rather retain memory from use and practice. Regardless a good example of something to remember in math is the quadratic formula Still enjoyable to derive and 'see' why it works but also just used so much. That said, I wouldn't encourage memorizing it without first understanding it.
    - PascLeRasc 8y ago
      The best is the 9's table, how all the digits add up to 9. My mom's an elementary teacher and there's always a few third graders who figure that out on their own and love it.
      - sdenton4 8y ago
        I was legit angry that I didn't learn about the divisibility checks for 3 and 11 until college...
      - somethingroma 8y ago
        Additionally 9 * n = [n-1, 10-n] for n = 2-11; where n-1 is the digit in the 10's place and 10-n in the single place. This just an aesthetic curiosity. I know the pattern continues for larger n I've just never bothered to generalize it. Also never compared it to other bases.
        Koshkin 8y ago
        > [n-1, 10-n] for n = 2-11 I think the usual 10 * n - n makes more sense and is much easier to remember.
        sdenton4 8y ago
        It's not an aesthetic curiosity at all! 10 is equal to 1 mod 9... So suppose X is written in base 10 (a0 * 10^0 + a1 * 10^1 + a2 * 10^3 +...) and you want to find X mod 9. Then all of the 10^k's are just 1 (mod 9), so you just get the sum of the digits. So if X is divisible by 9, then the sum of the digits (mod 9) is zero. Same works for 3 (x is div by 3 iff the sum of the digits id divisible by 3). And 11 gets an /alternating/ sum of the digits, since 10 is -1 mod 11...
    - aj7 8y ago
      No. If you can’t multiply in your head, you are crippled in any quantitative reasoning. You have interjected too many steps in estimating, calculating, judging, etc. This is not to say there aren’t useful, non-quantitative pursuits.
  - OskarS 8y ago
    I don't think that it's "memorization" that's the key: I think it's repetition. Learning anything, but especially mathematics, requires you to just sit down and solve problems over and over again. That is what builds the right connections in your mind, gives you a feel for the shape of the solution, and gives you good intuitions about problems. Like: it's a pretty useless skill to have memorized what the anti-derivative of the cosecant function is. But if you solve enough integrals, and are familiar with the techniques, you can see the path to the solution of the problem, even without having memorized every single function and anti-derivative.
  - ummonk 8y ago
    You can just do a lot of multiplications, and let your brain cache the results over time.
    - rambl30n 8y ago
      This. Although said tongue-in-cheek, is a basic principle. Once you understand the concept and the mechanics behind something, what rote learning buys you is to free cycles in the future so that you can take up the next-level task. Skipping this step is like expecting someone to play the piano just by learning how to read a music sheet and where each note is in the keyboard. You build up your muscle memory so that you're able to take on more complex pieces as you progress.
    - Koshkin 8y ago
      Spending time memorizing the multiplication table, then, is more efficient on both accounts: you will not need to do "a lot of multiplications" to begin with, and it takes less time for your brain to "cache the results".
      - abecedarius 8y ago
        Maybe that works best for some people, but as a kid I didn't do it. I used a printed multiplication table while tackling some more-interesting problems[0], and let the table soak in as a byproduct. It went quickly and did not turn me off on math for life. Paul Lockhart in Arithmetic also recommended this. [0]: Multiplying two-digit numbers by one-digit, and that sort of thing. Lockhart had more artistic pursuits in mind. (An obvious reason this might not apply: I was more talented than my grade-school peers. But most kids would learn arithmetic better if not forced to before they're ready, and then the talent difference would matter less.)
  - credit_guy 8y ago
    May I add my support to your opinion? Or, to put it more bluntly, the "why" is overrated. The "modern" math education (in the US) comes courtesy of people like Jean Piaget or Seymour Papert. These guys were true geniuses, but made a grave error. They generalized from their own experience ("to understand is to invent") to everyone else. Unfortunately, for 99% of the people, the style of learning by discovery is extremely inefficient. So we ended up with a system that sounds great in theory, but in practice you have kids in the fourth of fifth grade still going over addition and multiplication.
rb808 8y ago
A good example of why well educated parents is more important than getting into the "the best schools".
- tesin 8y ago
  Isn't that just a matter of chicken and the egg timing tautology? The parents are well educated presumably because they went to good schools.
  - gaius 8y ago
    Not necessarily - many parents highly value education and encourage their kids despite and/or because they don’t have much themselves. Recent immigrants are a good example of this phenomenon.
    - tesin 8y ago
      I don't disagree with that statement, but rb808 specifically cited "well educated parents".
  - saagarjha 8y ago
    No, I think you’re getting the relation backwards. Many parents who were well educated have the opportunities to send their children to good schools, regardless of whether they went to a good school themselves, because they realize that it makes it easier to be well educated there.
    - tesin 8y ago
      How would someone be well educated at a bad school? I feel like that loses any reasonable definition of good.
      - saagarjha 8y ago
        A school's "rating" is merely an average of the quality of education students there receive. It's entirely possible to "get lucky" and have an exceptional teacher or participate in an optional extracurricular that helps them learn more than their peers. And, of course, there's also the chance that the student did well despite going to a bad school, by self-studying.
ivan_ah 8y ago
Yeah, private tutoring is the best. The custom, just-in-time curriculum that a tutor can prepare is the most efficient way to learn. I think equally important is the transmission of a relaxed, exploratory attitude towards the STEM subjects. When a learner sees their teacher take on positive attitudes towards complexity, they too develop this attitude and can go on to solve problems on their own. Unfortunately, this transfer of attitude works in the negative direction too. If the teacher is out of their comfort zone (e.g. parent touching math for the first time in 20 years), then the learner might pick up on the anxiety and start to consider the subject to be unpleasant or hard---if your parent was stressing out about helping you with algebra, then algebra must be a thing of formidable complexity! As another example, you can think of a teacher bored with the subject (e.g. prof who is teaching this course for the 17th time this semester) who will then transmit "this is stupid shit you have to know, because you're forced to take this course" attitude and students will pick up on this too... Luckily these days there are really good resources (youtube, free books, interactive demos) so hopefully we'll have more STEM people in the future. We won't make 100% of the population into STEM-experts, but it's not unrealistic to hope that everyone can become at least STEM-literate. I'd like to believe that I contribute to this with my books. This one in particular would be good for adult readers who want to rekindle their relationship with the subject: https://www.amazon.com/dp/099200103X/noBSmath https://www.amazon.com/dp/099200103X/noBSmath
- chrisweekly 8y ago
  Great post. FWIW in discussing this topic I've heard and now adopted the term "(in)numerate" vs "STEM-(il)literate" -- though the latter might be intended to be broader in scope?
  - Zeebrommer 8y ago
    I don't think it's very helpful to make such a distinction between people who "get it", and those who don't. It's a spectrum. Unless of course when it's relating to a disorder, like dyslexia for example.
  - tsumnia 8y ago
    As Zeebrommer mentioned, I'd steer clear of this thinking. Carol Dweck's entire research has been on "fixed" and "growth" mindsets, and their impact on people's performance. Instructors that believe someone is or is not an "X person" have a negative impact on students. That is, if I believe in the idea that there are STEM-illiterate and STEM-literature people, why should I both helping someone who is STEM-illiterate? Instead, adopting a growth mindset, that while one may not know something now, with time and effort they can gain understanding is much more powerful. As I've argued in other threads, this may include doing things that are frustrating or otherwise "not fun" to establish grit/perseverance. Think of it this way, when you were a baby, you were "spoon"-illiterate. If you parents decided you're just "not a spoon person", you would be raised not knowing how to use a tool everyone else thinks is easy. If you grew up never using a spoon, and it became a requirement for future career advancement - you might become frustrated because you were taught you "were not a spoon person" instead of "I need to learn how to spoon!"
gnode 8y ago
I think understanding the why of mathematics makes all the difference between forgetting it soon after an exam, and being able to use it later in life. I sometimes find myself needing to return to something I once knew in high-school / university to solve some problem, not being able to remember the specific methods, but the underlying bigger picture of the concepts comes back to me, and I can make progress from there.
bitlax 8y ago
Feynman on why questions: https://www.youtube.com/watch?v=36GT2zI8lVA https://www.youtube.com/watch?v=36GT2zI8lVA
unpythonic 8y ago
As much as the Common Core standards are criticized here in California, I have to say that the emphasis on the "why" behind every operation is really fantastic. It's usually in the last section of each homework, but it provides a good discussion point when going over the homework each night. Asking students to answer questions like the following are easy to zip through, but they provide a good place to pause and find a way to connect the physical mechanics of a solution to the reasoning behind it: "How would you explain to someone else why the fraction 1/2 greater than 1/4?" or "Why doesn't angle-angle-angle show congruence but angle-side-angle does?" I've also noticed that the Common Core brings in advanced topics earlier without announcing to the student that it is an advanced topic. Ideas from algebra are brought in at natural points of the discussion rather than making a big deal of it. By the time that they realize they're learning algebra, they are already into many of the "rules" that would have otherwise been taught by rote. If you understand why you have to multiply both sides 5 to find out x/5 = 30, it feels much less arbitrary when the rules are made more explicit later.
- sdenton4 8y ago
  I haven't worked in an american classroom, but as a math educator, thought the CommonCore looked great when it was first rolling out. Math education is (kind of strangely) a bit of a battlefield in the US (maybe not so strangely, when everything else seems to be, too)... I think a big part of it is prevalent math anxiety - perhaps embarrassment at seeing unfamiliar things on the kid's homework. Along with this, there's still lingering bad PR from the 'New Math' of the seventies, which made it somehow acceptable to make the argument that we should stick to teaching math the same way forever. (Especially for people who believe that math hasn't changed since Newton.)
  - zrobotics 8y ago
    Agreed, I can't stand the way we teach math in the US. I love math enough to have accidentally gotten a minor in it, but especially high school does nothing but train kids to hate math. My calculus class in high school was a net negative, since teaching to the test just caused many of my friends to switch away from STEM majors in college (if Calc 1 was that horrible, why become an engineer ahd have to suffer even more?). One of these friends went on to become a forensic accountant, so she obviously couldn't hate math that much, but that calculus class was traumatic enough that she summarily dismissed a biology major in college. I think one of the main pain points, which my dad was able to tutor me through, was understanding why. Learning calculus (or any math, for that matter) as just a list of mechanical steps is awful, whereas learning why we perform the mechanical steps allows one to glimpse the beauty behind math. I wonder how many children we've ruined this beautiful subject for?
jordigh 8y ago
> A ball is thrown off a building at a speed of 15m/s and at 30 degrees to the horizontal. If the building is 100m tall, how far from the base of the building will the ball land? g=9.8m/s² 280.17 metres, right?
- deleted 8y ago
  [deleted]
- Koshkin 8y ago
  Depends on whether the angle is above or below the horizontal. (The ball won't make it this far regardless.)
  - noobermin 8y ago
    That's a good point. I'm not one for details and I just assumed it above the horizontal because it makes it a touch more interesting.
- abdullahkhalids 8y ago
  Nope. That's too far. I am too lazy to write out the equations, so we can use an app such as this http://www.walter-fendt.de/html5/phen/projectile_en.htm http://www.walter-fendt.de/html5/phen/projectile_en.htm to find out that it falls 69.5 m from the base of the building.
  - jordigh 8y ago
    It's not so difficult to fix. -(9.8/2) t^2 + 7.5 t + 100 = 0 Positive root is 5.3472 seconds, multiply that by 15*sqrt(3)/2 to get 69.462
j2kun 8y ago
> Or why his first-generation ABC (“American-born Chinese”) children were more interested in sports or the humanities than studying fractions and common denominators. This could have just as easily been written: "Or why his first-generation ABC children were more interested in putting a ball in a hoop and learning languages that nobody speaks anymore than in noticing and thinking about the patterns and structure in the world around them." Framing the same underlying idea can disparage one thing while elevating another.
- ssivark 8y ago
  More importantly, it gives the reader a keen insight into the father's perspective -- which the author is trying to convey, and not many people would be familiar with.

172 points by danso 8y ago

"This was the key. If you knew hhhhwwhy you didn’t have to memorize equations, or solve equations in the exact same way they did in the book..." I feel like this is a valuable lesson a lot of people could benefit from.

tesin 8y ago
I agree as a generalization, but that does presume that the syllabus itself doesn't need rote interpretation. “No, we haven’t learned about some of the vectors yet, so for the sake of the problem, we just take this one out,” Many many students self-combust when faced with garbage worded questions because they never learnt how to deconstruct them in class. When questions come with spelling errors and ambiguous variables it's tough.
Koshkin 8y ago
Knowing "why" is exteremely helpful in many cases. But let's not forget that, on the other hand, memorization plays a huge part in learning, and not only in mathematics. Learning "why" multiplication works will not help you to retain the multiplication table in your head. (Also, in many cases the answer to a question "why" you might get will be plain wrong - sometimes because the correct answer is too complicated, sometimes for other reasons.)
- somethingroma 8y ago
  Memorizing multiplication tables is over rated. Seeing connections between those early multiples up to 12 is more interesting. Can be helpful I suppose in factorization. Maybe I just don't trust my memory (as in how am I sure that 7*8 is 56) and why I despise rote memorization and rather retain memory from use and practice. Regardless a good example of something to remember in math is the quadratic formula Still enjoyable to derive and 'see' why it works but also just used so much. That said, I wouldn't encourage memorizing it without first understanding it.
  - PascLeRasc 8y ago
    The best is the 9's table, how all the digits add up to 9. My mom's an elementary teacher and there's always a few third graders who figure that out on their own and love it.
    - sdenton4 8y ago
      I was legit angry that I didn't learn about the divisibility checks for 3 and 11 until college...
    - somethingroma 8y ago
      Additionally 9 * n = [n-1, 10-n] for n = 2-11; where n-1 is the digit in the 10's place and 10-n in the single place. This just an aesthetic curiosity. I know the pattern continues for larger n I've just never bothered to generalize it. Also never compared it to other bases.
      - Koshkin 8y ago
        > [n-1, 10-n] for n = 2-11 I think the usual 10 * n - n makes more sense and is much easier to remember.
      - sdenton4 8y ago
        It's not an aesthetic curiosity at all! 10 is equal to 1 mod 9... So suppose X is written in base 10 (a0 * 10^0 + a1 * 10^1 + a2 * 10^3 +...) and you want to find X mod 9. Then all of the 10^k's are just 1 (mod 9), so you just get the sum of the digits. So if X is divisible by 9, then the sum of the digits (mod 9) is zero. Same works for 3 (x is div by 3 iff the sum of the digits id divisible by 3). And 11 gets an /alternating/ sum of the digits, since 10 is -1 mod 11...
  - aj7 8y ago
    No. If you can’t multiply in your head, you are crippled in any quantitative reasoning. You have interjected too many steps in estimating, calculating, judging, etc. This is not to say there aren’t useful, non-quantitative pursuits.
- OskarS 8y ago
  I don't think that it's "memorization" that's the key: I think it's repetition. Learning anything, but especially mathematics, requires you to just sit down and solve problems over and over again. That is what builds the right connections in your mind, gives you a feel for the shape of the solution, and gives you good intuitions about problems. Like: it's a pretty useless skill to have memorized what the anti-derivative of the cosecant function is. But if you solve enough integrals, and are familiar with the techniques, you can see the path to the solution of the problem, even without having memorized every single function and anti-derivative.
- ummonk 8y ago
  You can just do a lot of multiplications, and let your brain cache the results over time.
  - rambl30n 8y ago
    This. Although said tongue-in-cheek, is a basic principle. Once you understand the concept and the mechanics behind something, what rote learning buys you is to free cycles in the future so that you can take up the next-level task. Skipping this step is like expecting someone to play the piano just by learning how to read a music sheet and where each note is in the keyboard. You build up your muscle memory so that you're able to take on more complex pieces as you progress.
  - Koshkin 8y ago
    Spending time memorizing the multiplication table, then, is more efficient on both accounts: you will not need to do "a lot of multiplications" to begin with, and it takes less time for your brain to "cache the results".
    - abecedarius 8y ago
      Maybe that works best for some people, but as a kid I didn't do it. I used a printed multiplication table while tackling some more-interesting problems[0], and let the table soak in as a byproduct. It went quickly and did not turn me off on math for life. Paul Lockhart in Arithmetic also recommended this. [0]: Multiplying two-digit numbers by one-digit, and that sort of thing. Lockhart had more artistic pursuits in mind. (An obvious reason this might not apply: I was more talented than my grade-school peers. But most kids would learn arithmetic better if not forced to before they're ready, and then the talent difference would matter less.)
- credit_guy 8y ago
  May I add my support to your opinion? Or, to put it more bluntly, the "why" is overrated. The "modern" math education (in the US) comes courtesy of people like Jean Piaget or Seymour Papert. These guys were true geniuses, but made a grave error. They generalized from their own experience ("to understand is to invent") to everyone else. Unfortunately, for 99% of the people, the style of learning by discovery is extremely inefficient. So we ended up with a system that sounds great in theory, but in practice you have kids in the fourth of fifth grade still going over addition and multiplication.

rb808 8y ago

A good example of why well educated parents is more important than getting into the "the best schools".

tesin 8y ago
Isn't that just a matter of chicken and the egg timing tautology? The parents are well educated presumably because they went to good schools.
- gaius 8y ago
  Not necessarily - many parents highly value education and encourage their kids despite and/or because they don’t have much themselves. Recent immigrants are a good example of this phenomenon.
  - tesin 8y ago
    I don't disagree with that statement, but rb808 specifically cited "well educated parents".
- saagarjha 8y ago
  No, I think you’re getting the relation backwards. Many parents who were well educated have the opportunities to send their children to good schools, regardless of whether they went to a good school themselves, because they realize that it makes it easier to be well educated there.
  - tesin 8y ago
    How would someone be well educated at a bad school? I feel like that loses any reasonable definition of good.
    - saagarjha 8y ago
      A school's "rating" is merely an average of the quality of education students there receive. It's entirely possible to "get lucky" and have an exceptional teacher or participate in an optional extracurricular that helps them learn more than their peers. And, of course, there's also the chance that the student did well despite going to a bad school, by self-studying.

ivan_ah 8y ago

Yeah, private tutoring is the best. The custom, just-in-time curriculum that a tutor can prepare is the most efficient way to learn. I think equally important is the transmission of a relaxed, exploratory attitude towards the STEM subjects. When a learner sees their teacher take on positive attitudes towards complexity, they too develop this attitude and can go on to solve problems on their own. Unfortunately, this transfer of attitude works in the negative direction too. If the teacher is out of their comfort zone (e.g. parent touching math for the first time in 20 years), then the learner might pick up on the anxiety and start to consider the subject to be unpleasant or hard---if your parent was stressing out about helping you with algebra, then algebra must be a thing of formidable complexity! As another example, you can think of a teacher bored with the subject (e.g. prof who is teaching this course for the 17th time this semester) who will then transmit "this is stupid shit you have to know, because you're forced to take this course" attitude and students will pick up on this too... Luckily these days there are really good resources (youtube, free books, interactive demos) so hopefully we'll have more STEM people in the future. We won't make 100% of the population into STEM-experts, but it's not unrealistic to hope that everyone can become at least STEM-literate. I'd like to believe that I contribute to this with my books. This one in particular would be good for adult readers who want to rekindle their relationship with the subject: https://www.amazon.com/dp/099200103X/noBSmath https://www.amazon.com/dp/099200103X/noBSmath

chrisweekly 8y ago
Great post. FWIW in discussing this topic I've heard and now adopted the term "(in)numerate" vs "STEM-(il)literate" -- though the latter might be intended to be broader in scope?
- Zeebrommer 8y ago
  I don't think it's very helpful to make such a distinction between people who "get it", and those who don't. It's a spectrum. Unless of course when it's relating to a disorder, like dyslexia for example.
- tsumnia 8y ago
  As Zeebrommer mentioned, I'd steer clear of this thinking. Carol Dweck's entire research has been on "fixed" and "growth" mindsets, and their impact on people's performance. Instructors that believe someone is or is not an "X person" have a negative impact on students. That is, if I believe in the idea that there are STEM-illiterate and STEM-literature people, why should I both helping someone who is STEM-illiterate? Instead, adopting a growth mindset, that while one may not know something now, with time and effort they can gain understanding is much more powerful. As I've argued in other threads, this may include doing things that are frustrating or otherwise "not fun" to establish grit/perseverance. Think of it this way, when you were a baby, you were "spoon"-illiterate. If you parents decided you're just "not a spoon person", you would be raised not knowing how to use a tool everyone else thinks is easy. If you grew up never using a spoon, and it became a requirement for future career advancement - you might become frustrated because you were taught you "were not a spoon person" instead of "I need to learn how to spoon!"

gnode 8y ago

I think understanding the why of mathematics makes all the difference between forgetting it soon after an exam, and being able to use it later in life. I sometimes find myself needing to return to something I once knew in high-school / university to solve some problem, not being able to remember the specific methods, but the underlying bigger picture of the concepts comes back to me, and I can make progress from there.

bitlax 8y ago

Feynman on why questions: https://www.youtube.com/watch?v=36GT2zI8lVA https://www.youtube.com/watch?v=36GT2zI8lVA

unpythonic 8y ago

As much as the Common Core standards are criticized here in California, I have to say that the emphasis on the "why" behind every operation is really fantastic. It's usually in the last section of each homework, but it provides a good discussion point when going over the homework each night. Asking students to answer questions like the following are easy to zip through, but they provide a good place to pause and find a way to connect the physical mechanics of a solution to the reasoning behind it: "How would you explain to someone else why the fraction 1/2 greater than 1/4?" or "Why doesn't angle-angle-angle show congruence but angle-side-angle does?" I've also noticed that the Common Core brings in advanced topics earlier without announcing to the student that it is an advanced topic. Ideas from algebra are brought in at natural points of the discussion rather than making a big deal of it. By the time that they realize they're learning algebra, they are already into many of the "rules" that would have otherwise been taught by rote. If you understand why you have to multiply both sides 5 to find out x/5 = 30, it feels much less arbitrary when the rules are made more explicit later.

sdenton4 8y ago
I haven't worked in an american classroom, but as a math educator, thought the CommonCore looked great when it was first rolling out. Math education is (kind of strangely) a bit of a battlefield in the US (maybe not so strangely, when everything else seems to be, too)... I think a big part of it is prevalent math anxiety - perhaps embarrassment at seeing unfamiliar things on the kid's homework. Along with this, there's still lingering bad PR from the 'New Math' of the seventies, which made it somehow acceptable to make the argument that we should stick to teaching math the same way forever. (Especially for people who believe that math hasn't changed since Newton.)
- zrobotics 8y ago
  Agreed, I can't stand the way we teach math in the US. I love math enough to have accidentally gotten a minor in it, but especially high school does nothing but train kids to hate math. My calculus class in high school was a net negative, since teaching to the test just caused many of my friends to switch away from STEM majors in college (if Calc 1 was that horrible, why become an engineer ahd have to suffer even more?). One of these friends went on to become a forensic accountant, so she obviously couldn't hate math that much, but that calculus class was traumatic enough that she summarily dismissed a biology major in college. I think one of the main pain points, which my dad was able to tutor me through, was understanding why. Learning calculus (or any math, for that matter) as just a list of mechanical steps is awful, whereas learning why we perform the mechanical steps allows one to glimpse the beauty behind math. I wonder how many children we've ruined this beautiful subject for?

jordigh 8y ago

> A ball is thrown off a building at a speed of 15m/s and at 30 degrees to the horizontal. If the building is 100m tall, how far from the base of the building will the ball land? g=9.8m/s² 280.17 metres, right?

deleted 8y ago
[deleted]
Koshkin 8y ago
Depends on whether the angle is above or below the horizontal. (The ball won't make it this far regardless.)
- noobermin 8y ago
  That's a good point. I'm not one for details and I just assumed it above the horizontal because it makes it a touch more interesting.
abdullahkhalids 8y ago
Nope. That's too far. I am too lazy to write out the equations, so we can use an app such as this http://www.walter-fendt.de/html5/phen/projectile_en.htm http://www.walter-fendt.de/html5/phen/projectile_en.htm to find out that it falls 69.5 m from the base of the building.
- jordigh 8y ago
  It's not so difficult to fix. -(9.8/2) t^2 + 7.5 t + 100 = 0 Positive root is 5.3472 seconds, multiply that by 15*sqrt(3)/2 to get 69.462

j2kun 8y ago

> Or why his first-generation ABC (“American-born Chinese”) children were more interested in sports or the humanities than studying fractions and common denominators. This could have just as easily been written: "Or why his first-generation ABC children were more interested in putting a ball in a hoop and learning languages that nobody speaks anymore than in noticing and thinking about the patterns and structure in the world around them." Framing the same underlying idea can disparage one thing while elevating another.

ssivark 8y ago
More importantly, it gives the reader a keen insight into the father's perspective -- which the author is trying to convey, and not many people would be familiar with.