In Math Cram Sessions, Solving for Why
nytimes.com
nytimes.com
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
Often they'll know that they know the answer, but they don't realise that they also know why it's the answer!
That's a fair bit above the regular rate though, you can get a decent tutor for less.
However, I wanted to pick up a couple of your examples - "parent touching math for the first time in 20 years" can also often be a great way for students to learn!
Adults are usually able to figure their way around a subject and this is a great learning point for kids - that you don't have to know everything, if you know how to find out about it.
Any parent has got through undergraduate education in any subject (and many who haven't) - who have learned how to learn - can find out what they need to know by looking it up and finding resources that help their comprehension. Being stressed about it is a different situation, and of course I agree with you there.
I also do sometimes say "This is stupid shit you have to know for the course" - because many courses contain bad content - but I will follow it up with "if we have time, it will be more practical to think about..." That also isn't a bad thing necessarily.
It's important for students' development to critically analyse why they are doing a particular course - if they're doing it to pass, then they should first pass.
As for teaching outside the comfort zone, I actually think that's a great opportunity. It puts you in the same place as your student so you can model the meta-cognitive skills of learning, curiosity, etc. You can even demonstrate how to 'fail' at something with grace - and give the student the opportunity to be better than their teacher at something!
I really disagree with your choice of words here, specifically, “just-in-time”. This makes it seem like the tutor is teaching things to the student right before they need it for an exam, which is an unfortunate effect I’ve seen happen because in the long run it doesn’t really teach anything. But where a private tutor really does excel is having a better understanding of how to tailor the curriculum to the student (this includes both simplifying or making it more challenging, depending on the student and the topic).
I was using “just-in-time” in the sense of just-in-time-for-the-lesson, e.g., if the tutor wants to teach concept X which requires knowledge of prerequisite concepts α1, α2, and α3, then the tutor can provide a quick review of these concepts before explaining X. Tutors can do this "custom filling in of gaps," whereas in a groups setting the teacher might be forced to say "you should know this already."
On the contrary, I found it really precise. JIT processes in manufacturing are about providing the correct inputs at the time they're needed. If your desire is mastery over the material, a teacher who can predict or extemporize tutorials on the subject as you need them is invaluable.
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!"
standard primary literature caveats etc and where it is/is not applicable but this does put 'growth mindset' on the back foot.
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.
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.)
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?
If you've got a child, I'd highly recommend reading through the standards in both math and ELA. They're well done, particularly the math.
We like to pretend that all kinds of jobs require knowing algebra and geometry, but no, but that is greatly exaggerated.
What you learn when you study those subjects that you have a decent chance of actually using on the job is how to reason.
There were a couple of interesting essays on this point by Underwood Dudley.
One is called "What is Mathematics For?" (which he admits is click-bait, and it should really be called "What is Mathematics Education For") from the May 2010 "Notices of the AMS" [1].
The other is called "Is Mathematics Necessary?" [2], which I believe is an earlier, shorter version of the what later became the AMS article.
The one reason why I dislike Common Core is that this kind of explanation get quickly become tedious if you already understand what’s going on and it’s “obvious” to you. And many teachers seem to like to use this section as an excuse to waste their students’ time at home by forcing to write much more than they really need to.
If that's the case, then it sounds like your child needs to be in a higher level math course. If they're already at the top, then I don't think having to explain in detail the "why" behind concepts will be a net negative for your child.
Learning how to explain a concept simply and succinctly is critical to most modern knowledge-based jobs.
Just curious, have you read all the way through the Common Core math standards? Most of the people I know in technical and engineering professions who have read completely through it (and not just a single section or grade level), come out with a lot of respect for the standards.
> If that's the case, then it sounds like your child needs to be in a higher level math course.
This isn't necessarily possible at school, and besides, even if it was, this may not be the best idea (for developmental, social, etc. reasons). Outside of school, sure–she is doing things to get ahead and learn more.
> If they're already at the top, then I don't think having to explain in detail the "why" behind concepts will be a net negative for your child.
This is exactly the group I think it's a net negative for: you're forcing them to put what they feel is "obvious" into words. This is fine once or twice, or when teaching others, but on assignments you have to do it repeatedly, and at some point you'll just give up and start writing things that are just different ways to word "because that's the next thing we have to do" to fill space.
> Just curious, have you read all the way through the Common Core math standards? Most of the people I know in technical and engineering professions who have read completely through it (and not just a single section or grade level), come out with a lot of respect for the standards.
No, I haven't, but I'm sure they're not bad. It's just that the way they ended up being implemented hasn't really impressed me, since it seems like the goals they've laid out aren't being achieved.
A good math student understands the subject in a principled way, which shapes inference and intuition for future discovery, and results in rock solid grasp of the subject.
Supposedly, this is the point of the Common Core math curriculum, right?
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.
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.
I feel like this is a valuable lesson a lot of people could benefit from.
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.
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.
This is not to say there aren’t useful, non-quantitative pursuits.
Incase this is what you mean: remembering the multiplication table is the antithesis of multiplying in ones head.
But it's not! Because in order to be able to multiply numbers in your head efficiently you must remember the multiplication table.
Additionally my experience tells me otherwise.
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...
I think the usual 10 * n - n makes more sense and is much easier to remember.
[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.)
“No, we haven’t learned about some of the vectors yet, so for the sake of the problem, we just take this one out,”
This became especially apparent during my first year of university where I took a macro economics course. Because I had already had a bit of calculus, so many of the mathematical problems in that course were almost trivial. Friends that didn't have any calculus found the course a lot more difficult because they had to memorize so many equations.
I have kids now and I watch them memorizing things and I can't help but think that's a waste of time. Remembering what the terms in the quadratic equation isn't nearly as important as remembering what the uses of the quadratic equation are. When do you deploy it? What does it give you? That's how I wish it was taught.
I think a good education is more about the student than the university. The reason for the strong correlation to the contrary is because good universities get to hand pick from those overachievers whom would achieve with or without said university. And that character development happens long before one's applying to universities.
That said a top university, its resources, and atmosphere of being surrounded by other top students, certainly helps foster even more success - but I expect if you take the average merit admitted MIT student and transplanted them into Party State U after 4 years you'd still find an extremely 'well educated' individual. By contrast, transplant your average Party State U student into MIT and you'd find a dropout.
Source?
280.17 metres, right?
-(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