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In Math Cram Sessions, Solving for Why (nytimes.com)
172 points by danso 10 days ago | hide | past | web | favorite | 80 comments





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


This is a minor point, but I found understanding the "discourse" behind a field super helpful. In abstract algebra my professor spent a lot of time why we study certain structures (groups, commutative rings, modules, fields) more than others. One time he was making this super frustrated remark that our book (Dummit&Foote) calls ring a non-unitary ring whereas in class he'll call ring as a ring with unity (i.e. a ring that has 1). He was frustrated because for him this was not an interesting distinction and "to have interesting results you need to introduce a 1 anyway". I think this sort of emotional and human responses emphasizes what is important to understand in that field and what is not. Some people saw algebra as a laundry list of structures with axioms, whereas it's just another way of solving problems and structure is the approach, so finding useful structures for the problem in hand is more important than studying all those structures, as if memorizing the periodic table.

Private maths tutor here — what I find most gratifying about tutoring is revealing to my students how much they already know.

Often they'll know that they know the answer, but they don't realise that they also know why it's the answer!


How much are your services? I'm an out-of-school adult that never really grasped math but hate that I don't know more about it to this day. Just wondering what kind of range I'd be looking in to just get some explanations or something. I don't currently have any specific questions though.

~150 AUD/hr for in-home

That's a fair bit above the regular rate though, you can get a decent tutor for less.


Have you tried Khan Academy? It lacks all the person interaction and customization of a private tutor but they have (IMHO) really good explanations and quite good problems to solve for all math through high school.

Love their resources, but having somebody to guide you to the resources that match your deficit is the real magic imo.

I'm a private tutor so obviously I agree it's the best ;-) thanks

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.


I think this transference of attitude is a really insightful model for teaching. One of the other methods I've seen used (especially for math) is treating problems like a game. There is a joy in solving even difficult problems, that you can teach gradually with the right feedback... and learn.

When I was 16/17 I did private maths tutoring for some of my classmates. As a tutor I enjoyed the conversations we had about 'why'. These people were missing a lot of context about why they were even doing anything, so the steps were all mechanical and it was hard for them to do anything beyond what they'd memorised. Conversations about the material, outside what's strictly required to solve a problem, really work wonders.

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!


> 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 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).


Ah no, my intended meaning of “just-in-time” was definitely not in the sense of just-in-time-for-the-exam. In fact I think there is very little use for last-minute cramming... at most you can survive the test, but definitely not build fluency or understanding.

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."


> This makes it seem like the tutor is teaching things to the student right before they need it for an exam

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.


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?

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!"


aside: Sisk et al (To What Extent and Under Which Circumstances Are Growth Mind-Sets Important to Academic Achievement? Two Meta-Analyses DOI 10.1177/0956797617739704) looked at >450,000 students and found little or no effect size for this approach.

standard primary literature caveats etc and where it is/is not applicable but this does put 'growth mindset' on the back foot.


Interesting find, I had no idea! I'll read through it, I'm curious if it discusses Dweck's "helpness" terminology as well.

I think you've misunderstood me or conflated my comment with someone else's; mine said nothing about fixed vs growth mindset. The term "innumerate" -- just like "illiterate" -- indicates the current state of a person's abilities, not some fixed or inherent trait. The remedy is education.

My apologies, this my interpretation of your comment. Thanks for the clarification

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.

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.


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.)


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?


I doubt the critics of CC know what "New Math" was.

There's so much misinformation on the Common Core that many people don't realize how much of a modest adjustment it was from before. Another way of summarizing the changes made by the Common Core, aside from your note that it focuses more on why, is that it advances the math curriculum by about half a year.

There is indeed so much misinformation! People should read the standards themselves, and also differentiate between the Common Core standards and the curricula being sold to districts.

Reading through the actual Common Core standard was what convinced me that it was a step in the right direction. It's clear that a lot of careful thought went into the Common Core math standards.

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.


One can make a good argument that the "why" of math is really the only reason to teach it as a standard subject required for everyone, at least beyond just elementary arithmetic.

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.

[1] https://www.ams.org/notices/201005/rtx100500608p.pdf

[2] http://www.public.iastate.edu/~aleand/dudley.html


> 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.

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.


> 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.

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.


I don't actually have a child: I was merely basing this observation off watching my sister go through Common Core and compare to my (pre-Common Core) experience.

> 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.


I don't know whether or not Common Core is the "right" solution for math education, but on face value it is.

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've noticed this too. Also teaching different ways of solving the same problem - I had to make sure kids didn't go back to solving the way they knew how already.

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.


> 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.


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.

"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.


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.)

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.


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.


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.


I don't understand what you're saying or objecting to.

Incase this is what you mean: remembering the multiplication table is the antithesis of multiplying in ones head.


> 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.


No. By definition remembering an answer is not the same as doing.

Additionally my experience tells me otherwise.


I'm sorry but this seems patently wrong to me. There's only so much working stack space in your brain. If you're constantly having to think through multiple steps to multiply single digits then you're going to be at a serious disadvantage when you need to solve a bigger problem that involves more work than just single-digit multiplication.

No remembering the table is the ROM-table part. You need the algorithm part too.

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.

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.

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...


> [n-1, 10-n] for n = 2-11

I think the usual 10 * n - n makes more sense and is much easier to remember.


I was legit angry that I didn't learn about the divisibility checks for 3 and 11 until college...

You can just do a lot of multiplications, and let your brain cache the results over time.

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.

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".

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.)


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.

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.

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

Uhm, no - asking students to prove theorems and lemmas and answer all the "whys", instead of just applying a set of problem solving rules was one of the main reasons why the high school I went to was so good at math education. Our physics teacher constantly told us not to memorize the formulas, but to understand the logic behind them so we could just figure them out if we needed them. It's a better way to learn math and physics indeed, but a good teacher can do it too.

I think the order related subjects are taught isn't always the best. Once I learned calculus, a lot of things I memorized in physics stopped being something I had to remember. Velocity is just dt/dt. Acceleration is just dv/dt. Give me a graph of velocities and I can find the distance traveled by integrating.

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.


Everyone should be well-versed in linear algebra and calculus before learning quantitative physics, economics, etc. It just makes the subjects so much easier.

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.

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.

I don't disagree with that statement, but rb808 specifically cited "well educated parents".

And so it's turtles, all the way down?

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.


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.

How would someone be well educated at a bad school? I feel like that loses any reasonable definition of good.

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.

the best teacher(s) in a 'bad' school overlap with the worst teachers in a 'good' school. Keep in mind that there is a relationship between student and teacher and that clicking with one teacher isn't guaranteed or that after a year of growing up, you might click with a teacher you didn't get on well with the year previous.

Aggregate test scores say more about the student/neighborhood population than the instruction. “Bad” schools become “good” as their neighborhoods gentrify, for example.

My parents both hated math but I loved it and excelled at it through public school. That was due 100% to my own interests and having two PhD math teachers in high school.

Thanks for this comment. I said something similar in another thread. Resources should be spent on making sure there are qualified teachers teaching math.

No. It's an example of why it is important to have parents devote time to helping their children study. This used to be much more common in the past.

>This used to be much more common in the past.

Source?



> 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?


Depends on whether the angle is above or below the horizontal. (The ball won't make it this far regardless.)

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.

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 to find out that it falls 69.5 m from the base of the building.

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




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