I never had any problems with math until I went to university, so I was merely a passive observer of everyday struggle for some people. I honestly believe that foundations are the key. Either you're taught to think critically, see patterns and focus on the train of thought, or you focus on numbers and memorization.
The latter obviously fails at some point, in many cases sufficiently late to make it really hard to go back and relearn everything.
Math is extremely hierarchical and I believe schools do not do enough to make sure students are on the same page. If we want to fix teaching math, I would start there, instead of working on motivation and general attitude. Those are consequences, not the reasons.
I indeed believe you are wrong. I started getting interested in STEM when I was 16 and now have a masters in math on the one hand and work for one of the most prestigious Tech firms in existence on the other. Meanwhile, when I was young, I was encouraged to enter creative fields (mostly writing in my case) which I did have enthusiasm for but then dropped as a teenager. So I switched at least twice, what I wanted to do with my life and still turned out fine.
Honestly, the whole "you need to get them when they are young" idea never convinced me very much. It's never too late to learn a thing and often you need a certain age to actually appreciate something. You don't have to be a childhood prodigy to be good at something.
I rather find the idea pretty harming; I've met a lot of people who did not pursue the career they wanted, because they bought into the myth, that you need to start programming at 6 to get a job at a company like Google or Facebook.
Let kids be kids for a while. Don't worry, they can always figure out what they love later and it's never too late to reconsider.
Yes, you get a word problem about Jack and Kwanzeeah trying to fill up a pail or something, it was nonsense to me. Why are they filling up a pail? Why are they using a bucket that is 173/9th's the size of the hose? Why do they have to not let it overflow? Why do I have to decipher this inanity, why can't they just ask me to do this stupid problem of stupid fractions? Why the cloak and dagger?
Then you get to 'real' algebra and you start to find roots of quadratic equations and other useless information. Dear Lord! When has anyone ever done this for any reason? Yes, something something eigenvalues something something computational modeling something runge-kutte. But for real though, quadratic equations are just nonsense gibberish to weed out the poor kids from the rich ones. Trig, yes, it's useful when you are building a shed that has to be totally perfect and crisp otherwise the rich white lady will not pay you. Geometry was a bit 'fun' actually, but in the same way that learning that Custer's Men took a poop at this road-side stop on the way to Little-Big Horn.
It wasn't until Chem in my 10th grade year that I actually 'got' algebra. We were doing moles to molarity calculations to get the solution to the right pH and make it all turn pink or something. It was then that I realized that all this mathy stuff was actually useful to me, that I could use it to make things easier and better for myself and my family. Like, I could 'know' what to do then. Before that it was just drill and kill and bad grades and shame. I remember staying after class and into the next one in the room, just sitting in the back doing these calculations over and over. I got kicked out for doing math! It was such a relief! I finally 'got it' when I needed it for something real to me personally.
For me, it was the issue of 'math' being this blunt-shame-thing that made no sense to use ever. But once I needed to use math, it was trivially easy. Learning math for each of us is individual, we each have our own motivations that are unique that need to be met. I know that is not easy to institutionalize, but if you want to do it, you have to find the 'correct motivation for each kid.
This is close to my experience as well. Math, for me, wasn't super difficult, but it didn't necessarily come naturally either, and I got really bored with the tedious, mechanical aspects of it in algebra, college algebra, trig, etc. Geometry and Calc I were mildly interesting, but I was handicapped a bit in Calc I because my basic algebra skills weren't as strong as they needed to be... because I found basic algebra to be mind-numbingly boring.
Fast forward 20+ years and I'm not working on re-learning some lower level math + leveling up on higher level calculus and linear algebra. This time, it seems easier (although I might not go so far as to say "trivial") and I attribute that largely to the fact that I'm much more motivated now. Now I have specific reasons for wanting to learn this math (mostly to do with machine learning and AI) and that seems to make all the difference in the world.
I'm tired of hearing about "take the derivative" and "perform the integral" (or whatever), and not quite understanding it. Heck, just what I wrote probably indicates I don't quite get it!
Also, my intuition on probabilities and statistics isn't there either, so I want to do something to fix that as well. So, when I finally get done - that's my next goal (after a bit of a break).
It isn't that I'm terrible at math, I just never used much beyond basic linear algebra after leaving high school. I never went much beyond high school: I got an associates from a tech school (worthless, since the school is long defunct), and I've taken a few community college courses, plus my online MOOC stuff that's more recent.
I'm not hurting career-wise; in fact, that's never been an issue as a software engineer (started when I was 18, and just kept going, earning more over time and having fun). I'm not doing bad right now, living in Phoenix. It ain't SV - but then again, it doesn't have the downsides of it, either (not that there aren't certain downsides).
Now - after having done MOOCs in ML/AI - I am finding that I need more understanding and intuition about calculus and other mathematical subjects, so I can really apply what I have learned, and understand it at a deeper level, and perhaps do more with it. I've given thought to going back for a BS and maybe an MS (not that I need it - I just want to do it).
Good luck to both of you; I hope it all works out great!
I'm now working...
In high school, if you can convince a single person that they need to know calculus, more power to you. 99% of people don't and it's literally a waste of time. Teach people finance though...suddenly we have math that people will know is important.
Why did I have to memorize all that shit if there's a simple way to derive the formula? Jesus.
"So, you still didn't have derivation in math class?"
We explained, that we will have it in the final month of the school year, if we are lucky. Her reaction was:
"But we kinda need it already. But I should be able to teach you that in 20 minutes."
And she did. And then she spent the rest of the lesson going through some old equations that she might have spent hours explaining: "Remember how long it took us to get to i.e: equation relating gass pressure and temperature? We always used that finicky helper variable. I think I even named it delta. In reality, you just derive this base form, and do a simple substitution."
Ok, my memory is a bit hazy.
But she went on and on, through stuff we have been ehm, deriving without calculus for hours, that took with calculus around 3-4 steps.
The one thing I really wished I'd have learned more of, younger, is stats. Knowing how basic probability works would have been fantastic - it took a lot of poker theory to really understand it in my bones.
Parrots have deeper understanding than was required of us.
I didn't even get a cracker.
I've been playing with computer graphics recently; I've understood trig for a long time, but this is the first time I've actually felt a need to understand matrices. Exploring applications on my own is the only way that I've really felt connected to math since maybe late grade school or early high school.
If you're splitting the tip and the tax, and everyone is pitching in $3 to cover the Birthday Girl, that's more complicated than a lot of those problems they use that nobody cares about.
(In fact, I think if you put all of those math problems in terms of fairness, you'd get more kids to pay attention. Nobody wants to get cheated, and that's what happens when you're bad at math and/or finance).
That's quite insightful! I can think of a few related approaches that could round it out, like frugality (eg. only buy the paint/bugspray/lumber/etc. you actually need for your project), and laziness (do the least amount of work possible that still gets the job done).
I don't think it matters what ethnicity or race you are, it's good to be able to actually get things the right length. And it's a lot easier to do it on paper than run around for half a day with a tape measure.
i.e a common problem such as calculating the optimal blend of fuel in a reactor is done based on solving a system of linear equations with constraints such as Loss on ignition, maximum allowable ash content, minimum specific energy etc.
Industrial design as well uses simultaneous equations extensively. Do you want to build your aircraft wing out of CFRP or a titanium alloy? One way is to use a constraint based selection method something like strength to weight ratio vs cost.
edit: I should add 99% of the time these are solved programmatically with the aid of a computer. You do not solve these problems "high school math style".
For what it's worth I also did better with literature and history than math when I was young. Also, I struggled to pass Calculus 2 but breezed through upper level CS courses. It's a very fuzzy thing.
Depends on what we are talking about. The post I was responding to was talking about "the first years of education" and with 16, we are way past that.
Yeah, I know that's not what you mean, but you're missing a point. The premise of the article is that US has a STEM problem. I merely suggested that they're allocating resources wrong to fix it. I didn't mean they should introduce rigor and discipline or increase hours - I said that the best way to achieve results in teaching math is to fix it early, when it's most commonly broken (in my experience).
It actually helps you if you decide to switch interests, because you got the basics right, which in this case is the ability to think in an abstract way.
If you don't agree with me, you should have provided arguments that math problems do not arise early in education system, but they are introduced later.
One last thought - timing _does_ matter. That's why almost every professional athlete starts as a child. The question is how much you gain for starting early and I agree, that it is usually unreasonable to put too much pressure on kids. Pressure is not necessary to fix the system as far as I see it, though.
People need good teachers at all ages. I'm glad his college professors helped him out enough to lead him to major in math.
On my side I was good at math but crashed in college, so much I avoided it, it's the reason I went into programming. Until 2 or 3 years ago I clicked again, I started to see through abstract math and now I'm back to math/phys because it's beautiful again.
If your buddy finds pleasure and can walk the path I'm not surprised he's still walking :)
Professional athletes are highly competitive. People don't hire basketball players to throw balls into hoops; they hire them to beat other basketball players. Unless you are one of the best humans at basketball, you are worthless as a basketball player.
In almost any profession, you do not need to be the best to have value.
If you knew it's not what they meant then why did you start off saying that's what they meant? If you thought they were missing the point why didn't you just say that?
While starting early is neither necessary nor sufficient for success, the question should be: do the benefits of starting early outweigh risks? Like the parent comment, I think they do, if done correctly and gently, but it should be decided case-by-case for each child. The only risk I can think of is: if you push too hard, it can have the opposite effect of making the child hate it. So, one has to do it with utmost sensitivity and by responding to cues and feedback from the pupil.
> The only risk I can think of is: if you push too hard, it can have the opposite effect of making the child hate it.
That seems like a weird "risk".
But I agree that the tendency of programmers and public to assume that if you already don't know a lot about computers, you are lost case and stand no chance is harmful. There is strong tendency to overstate how difficult things are and that makes people look elsewhere.
Seeing an integral sign and understanding what it means, for example, can help simplify seemingly complicated mathematical expressions and make them easy to understand.
Later the curriculum only expands on these problems so it's even harder and harder to catch up.
I have been working with college freshmen struggling with basic linear equations and such. While the profession is marginally unrelated to Math (e.g. Graphical Designer), I still wonder to this day how people can have 20-30-40 years and not know how they could (with a calculator in their hand!!) calculate the price after the discount. How can someone feel ok with this?
Some of them get behind in one or more subjects (math, say) as early as 1st or 2nd grade and all their later teachers are stuck trying to give them the remedial math help the desperately need, while teaching their other students on-grade-level material, and also trying to teach the remedial kids enough of the new stuff that their scores on that year's standardized test don't land the teacher in hot water (though the kids won't get it at all). The result is that the kids fall farther behind every year but keep failing up to the next grade. They'll enter e.g. 6th grade with ~Dec. of 3rd grade math skills (more often than not their other subjects aren't much farther along, though not always). If they get an LD diagnosis they'll get some extra help but by then it's too late.
It sucks for all concerned.
Yearly age-to-competency distinctions continue by sheer force of tradition, and are harmful to all but the totally mythological "average" student.
Further, socially-speaking, being with your actual peers is extremely valuable, obviously.
But yes, the ideal would be a tailored-to-each-child education. Personally, having seen the power of a good Montessori education, I think it's frickin' genius and should be the template for all education (specifically the grouping of 3 yrs together -- e.g. 3,4,5-year-olds together), so kids are perpetually working through the cycle of "look up to someone, mimic someone, mentor someone".
... but I digress.
Fact is, grouping by age works socially and it's much, much cheaper than a tailored education, which is all we an afford (get taxpayers to pay for).
At some point, I expect a software company to make headway in this space and you'd see a bunch of kids staring at iPads all day and a bored teacher playing Minecraft at the front of the room, only engaging when someone gets stuck.
The math program I use with my kids is this one: http://www.defimath.ca/ecole-maison/ (in French)
It was developed by two mathematicians who actually studied in classrooms what worked on kids when teaching math. They've come up with a method where kids learn negative numbers and multiplication _before_ learning positional numeration (numbers greater than 10, with units, tens, hundreds, thousands, etc.). I've seen first-hand how these concepts just click in my daughter's mind, and how she often comes up with the new concept herself when you introduce the prerequisites in an order that makes sense to her.
Personally, I think Khan Academy is one of the few forward-thinking organizations doing something right in education. It tries to skew learning towards a 'mastery-based' model which keeps kids from falling behind by not advancing them too early. Of course, teachers and parents still need to implement the usage of Khan Academy in this way for it to be effective.
I would disagree... I think that there is huge value to being around a variety of people, and a strong negative value to only being around people in the same phase of life as yourself, especially when you are in the nasty phase. I think I made it through that part of my life because I had an after-school job where I fixed computers for a local office. I even enrolled in the vocational program at my highschool that let me out of school early to go to work (except every second Monday, when we learned how to spot shoplifters, short-change scams, and other hazards of the retail life.)
I think the number of people who felt good about their social lives, the things they did socially and the things, socially that were done to them in high school is... small. As far as I can tell, for most people, college is important because it is a kind of recovery from this, and prepares you for a workplace where conflict is muted, where yes, if you are good enough, you can still be an asshole, but where being an asshole has a pretty heavy cost that must be made up in other ways.
For me? Being expected to behave like an adult while being treated like an adult around a bunch of adults was amazing. It gave me a reason to keep going in high school, and when I came of age, I was all set to get a really nice job.
Sounds quite logical to me. If we assume a gaussian distribution (which tests seem to verify), most kids of the same age will have the same skills/level.
So at worse you mismatch what's taught to some kids towards the edges, whose level there are ways to accommodate anyway in most school systems (skipping a class or two for extra smart kids, or staying behind/supplementary teaching for less than average smarts).
And it's not just about learning and who can cram more into one's head (akin to e.g. preparation for the Olympics), but also about sharing the same teenager and adult-making experiences as other kids of your age, which is probably even more important that what's actually taught (the majority of which most people will forget anyway).
In this context, Gaussian is a pretty useless assumption without fixing a variance. Proposed alternatives range from "already implemented" to "totally infeasible" depending on variance.
> which tests seem to verify
Not really. For each individual subject area, maybe, and again, Gaussian is pretty uninformative.
But the odds of a student being "average" in every subject area != the odds of a student being "average" in a given subject area.
> whose level there are ways to accommodate anyway in most school systems
Except the whole point is that there are not currently ways to accommodate this in most school systems! From GP:
>> Schools (at least around here) do not hold kids back anymore. No matter what.
Also notice that holding back a student in math is possibly net detrimental if the student is not also behind in English and Science.
> but also about sharing the same teenager and adult-making experiences as other kids of your age, which is probably even more important that what's actually taught
Again, the prepubescent/tween/teen division is much less granular/restrictive than the competency-by-age-X division.
That doesn't really help us to know if the top achievers are being held back, for example. Perhaps we can speculate that lower achievers are being pulled up?
Outside school you're almost always going to be in age diverse groups; I think a larger element of that in public schools would be better. IMO it helps to emphasise that children are there to make their own education and not simply to be part of an age defined peer group where it appears you're doing something just because of your age, not because of the educational opportunity.
Retained students (the current terminology for "held back") are much more likely to stop attending school early, and on average do not see larger academic gains in their year retained than their equivalent peers who are not retained.
The ONLY effective intervention is dramatically increased services for that student, delivered rapidly once the deficiency is detected. Many high performing charter school networks actually use this systematically, but only by overworking their teachers and generally burning them out within 5 years (retention rates at 5 years are frequently <10%, compared to 50% in the broader profession). Most teachers in mainstream district schools are covered by union contracts that limit their required working hours, so for the vast majority of schools even this isn't an option. Actually delivering these services in a sustainable manner is a financial burden American taxpayers appear largely unwilling to shoulder.
It is the discount they care about, not the price. These have no calculations, only gut feelings, which are perfectly exploited.
Btw, how many world-ruling decisions are made this way?
One of my children showed natural talent in language at 9 months with no prompting. This was brought to our attention by childcare staff. Another of our children showed a natural talent with mathematical concepts at about a year old.
Even as they grew, our linguist struggled with math (for years) and our procedurally oriented child struggled with language (for years).
To this day, these two children maintain these core differences. It took at least 6 years for our linguist to crack basic arithmetic (even basic addition) which was at least several years behind our proceduralist.
I found out later that some leading child psychologists recognise different brain types in very young children (exactly as I found). An Internet search on brain types of children will show some high profile child psychologists who talk about this in depth, despite some strong "opinions" (ie. devoid of evidence) that oppose these studies.
Our linguist, with minimal pressure, has developed into a strong mathematician (at least grades wise) but to this day has never demonstrated anywhere near the natural ability of our proceduralist.
I have drawn the same conclusions with my own siblings and my wife's siblings. At a very young age, our own strengths become apparent without intervention. I am very glad I never pushed my kids to be equal (or even close to equal) in all skills. I consider most uses of the word "equal" worrisome (except for equal opportunity, a concept frequently downplayed in the last decade or so. Even Zuckerberg's famous open letter was unclear on such a fundamental concept).
OBS: when I asked our linguist to step through basic math, they understood the concept but could not do the work independently. Someone in this thread described a similar story for their child and attributed this to a lack of "confidence". For my child, I wholly reject that it was confidence related. When things clicked for our linguist, they clicked. If anything, our linguist's ability to crack the basics took patience on my part. I wanted my child to succeed quickly but I restrained myself (thankfully).
People need to realise that not all kids are the same. We have innate strengths. We have different learning styles, different learning rates, and different interests and motivations. I strongly reject the modern populist theory that we are all equal in ability and I believe we do significant harm because of this factoid. The motives behind this factoid concern me deeply.
If I could offer one piece of advice, your child(ren) are unique. Don't ever let anybody tell you that your child's strength or weakness comes from social conditioning. The only social conditioning cones from extreme behaviour (eg. Heavy handed forcing of "equality" under the banner of political correctness is extremely harmful, rather than focussing on potential and opportunity. This heavy handedness is also driving some extremely destructive social engineering under the banner of "equality". If you are watching academic trends you should be horrified as a patent).
I always encourage(d) play at a young age (physical activity, math games, language games). However, if you make this more than games (if you call this teaching and you start to measure), you set kids up for failure, especially when many children need patience and time.
According to PISA rankings, most western countries (especially English speaking) are not the top performers. I have hinted my beliefs of the root cause of this in this post. I predict most western countries will slip in ranking even further (especially English speaking countries). If things play as I expect, the slip will be significant in the next 10 years.
Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi without first explaining how it's calculated? Where does the "four-thirds" part come from in calculating the volume of a sphere?
Not to mention that there's never enough history to go along with the mechanics - who discovered the quadratic formula? What was their life like? Why were they playing with quadratic equations in the first place? This make math seem less like magic incantations and more like something that was sort of cobbled together by flawed weirdos in order to solve real-life problems, and evolved over time.
imagine a kid learning to speak 'stop just memorizing words, you need to understand how language was derived before you learn how to speak it'.
its completely counter intuitive to how creatures learn. learn easy things, especially those that relate to problems we deal with, and then get deeper into the subject if necessary.
Formulae without context are meaningless to most.
memorization is a side-effect of these, not a foundation.
If we're waiting until children are capable to derive those equations before we get them to use those a bit, then we're waiting for very long. And if you just keep telling them to add numbers and multiply numbers together for many years in a row without giving them any interesting problems to go with it, there'll be no one interested in math by the time they'd usually have the abstract maturity to deal with the more foundational modern math problems.
We don't ask Computer Science students to write an OS before using one, we don't ask carpentry students to build a hammer and cast nails before using them.
"hey, that's really neat! how does it work?"
"oh, you'll learn that in calculus, which we won't allow you to take for another four years."
Step 1: move pieces around and divide by the leading coefficient to put equation into the form x² + b = 2ax (or if you like, x² – 2ax + b = 0; or feel free to swap the sign of b if you prefer). The equation for the parabola is then x² + b = 2ax + y.
Step 2 (optional): rearrange that equation to get (x – a)² = a² – b
Step 3: x = a ± √(a² – b)
In this form, the “discriminant” is just a² – b, the x coordinate of the vertex is a and the y coordinate is b – a², Viète’s formulas tell us that the two roots satisfy ½(x₁ + x₂) = a and x₁x₂ = b. If the coefficients are real but the roots are complex, then we know each root has amplitude √b and phase arccos(a/√b). Etc.
This was tried. It was called "New Math". Spectacular failure. Do you want to know what worked? Memorizing times tables.
There's just no way around the fact that drilling is key to early mathematical learning.
I hate to admit it, but I am now a believer that strong arithmetic skills are important, and drills get you there. I don't like to call it memorization, since I'm not sure it necessarily is simply memorization. But you do need the answers at a moments thought. It should be as natural as saying a word.
And its not to say that you don't teach concepts concurrently... but that the arithmetic is fundamental.
That said, I still believe that the long division algorithm taught isn't so useful. :-)
Its hard to explain, or even to grasp, how fundamental basic arithmetic is to almost everything in math.
I disagree. Not hard to explain at all. Point the person at a page of geometry problems and say, "Imagine struggling with 90 + 90 at the same time as you struggle with the concepts here." Point the person at a page of polynomials to factor and say, "Imagine trying to do these if you hadn't memorized basic multiplication facts." And so on.
I still believe that the long division algorithm taught isn't so useful.
At the risk of not knowing precisely which algorithm you're talking about, I can't imagine one that doesn't work by taking a large/hard division problem and breaking it down into small/easy division problems. And that's useful because it's a great example of what math does for your thinking.
To be specific, I think the important principle math teaches is that, when faced with a big, hard problem, break it down into smaller, easier problems. I would rather describe math as a "learn how to break down problems" discipline rather than use the vague and pretentious "learn how to think" description. All areas of education help your thinking.
If you ask me 9 * 7 I still do 10 * 7 = 70 - 7 = 63 in my head; but, I seem to handle spectral graph theory just fine.
EDIT: As pointed out, I mistook OPs invocation of New Math to be talking about the much maligned Common Core rethinking of math education. I was briefly a high school math teacher, but before the roll out of these changes, so I can't comment first-hand on what the new curriculum looks like in the actual teaching. But I do know how poorly prepared my students were for math beyond arithmetic. They were trained with similar curriculum that I had experienced growing up in the 90s, which I think is poorly thought out.
Apologies for the confusion caused by me not recognizing the term New Math.
Concepts over computation (or well, before computation) is probably right. But New math was about learning the abstract before the concrete. This was predictably an abject failure.
I think the best way to teach math is to follow the trajectory that humanity took when discovering it. The key that's missing is that math doesn't just come out of thin air, its just a systematization of precise quantitative thinking. If we motivate the concepts using real world examples, then explain how to abstract away the particulars into a general procedure, then these connections will get made that make math "real" and relevant.
BTW, "not even wrong" is a rather rude way to point out a misunderstanding.
I find it disheartening that I have to teach my kids basic fractions, ratios, and transformation cause the teachers don't or barely touch on it. Kids are supposed to "discover" and "explore" math, whatever that means. In my opinion it's all bullshit.
Math, in many ways, is like an engine, either it works meaning the answer is correct, or it doesn't.
The way people are teaching math in the US is baffling and weird, but Pearson's textbook design has little to do with the common core per se. Most common core-labelled material is from something else with a new shiny CC cover slapped on.
Concepts of what? Without intuitive understanding of basic computation procedures, what concepts can anyone build out of nothing?
Math is concrete. It's from observation of actual quantitative phenomenon.
Modeling the world is a job that math the tool was created for, but it certainly isn't concrete just because the world that needed it was or is.
To split them, and think that, just because there is abstraction, and it's OK to develop the concept without the concrete substrates of actual experience is trying to make dream come true.
And I argue that no one should live in a dream. And it's obvious that only those dreams that have a strong connection with the real world have realistic chance of being made real.
Many problems that people have with math seem to stem from not having internalized the most basic facts about addition and multiplication. If you don't know at least to 10 by 10, each tiny step of working through a problem will tend to be interrupted by counting. Fluency requires memorization.
Also I never completely learned my multiplication tables to the point where they were reflexive, such was my loathing for rote memorization. To this day I sometimes need to pause to mentally crunch something. This sucks for small numbers but it means I have the mental tools to grind out bigger ones that other people would need calculators for.
People who were taught by memorising go into outrage a new type of exercise is introduced. Suddenly thinking is needed and that is bad in their eyes. Not exactly success. Meanwhile, you can memorize time tables in later age if you decide it is useful (people rarely do).
Once people see what they need out of mathematics for success in life, they never choose to memorize multiplication tables. But they do often learn new problem solving techniques.
That should tell us something about which is more important.
(But of course, we don't have to choose between the two either.)
The next steps are to start doing things backwards, "how many times do we count by four to get to twenty?" And we start to introduce notation. Bingo, simple division. This leads directly to simple fractions. This opens up conversations on adding and subtracting fractions, then multiplying and dividing them.
My oldest kid, now in college, could add, subtract, multiply, and divide fractions by second grade and understood them. Oddly, she had a horrendous time learning decimals. Her mental model of numbers was fractions and decimals were "weird." She would have to change things like 5.045 to 5 45/1000 to understand it, and wanted to work with it as a fraction. It took a long time for her to get comfortable working with decimals.
One time that I think it paid off. I told her that 0.999... is equal to 1. She said false. I said, no, it is true. Can you tell me why? At this time, she was in algebra, and I was expecting to show her how to prove it using algebra. She had a much better way of looking at it. In about a couple of seconds, she said, "well, 1/9 is 0.111... and 9/9 would be 0.999... and that is also 1." Her answer was much better than mine. :)
An example of when memorizing is bad (ie, when the underlying knowledge is skipped) was her 7th grade algebra teacher. In teaching the laws of exponents, he said "anything to the 1st power it itself and anything to the 0th power is 1. We don't know why, it is just one of those math things." Teaching like this is why we have students who, later in high school, can't do x^0 or x^1 because they think, "it is either 1 or 0 or itself, I don't remember." As opposed to applying mental models and patters to see that 3^3 -> 3^2 -> 3^1 -> 3^0 is just dividing by 3 each time. These students know 3^2 is 9. So they should know that the next is 9/3 = 3 and that the next is 3/3 => 1.
2) Did New Math fail because of poor results or because of popular revolt? (Would New Coke have failed if there was never any such thing as the original Coke?)
2a) If New Math did actually have poor results, was it because it hewed too closely to the goals I brought up, or because of other issues?
3) Memorizing times tables clearly didn't work, or we wouldn't be having this discussion
4) I never said that drilling times tables isn't important, so I'm not sure what your last sentence is in response to
I was told to memorize the times table, and tried but never managed to succeed. Instead, I found that I got along just as well by memoizing them instead; that is, I would compute the parts I needed on the fly in the margins of the paper. (Example: Say I need to find 37. I happen to know that 33=9, which I can double to get 36=18, plus 3 to get 37=21. These figures would be written down, so when I later needed 47 I could easily add another 7 to get 28. I had similar tricks for various other numbers, and could generally get the figure I needed--if it wasn't already written down--in a few hops.)
These contortions don't seem to have significantly affected my mathematical development, but they did* improve my logic and reasoning skills (or possibly merely showed that I had them). Particularly now that nearly everyone I know carries a calculator in their pocket, I don't see why we would continue to focus on rote learning over actually understanding how the underlying principles work.
I still feel a certain tinge of guilt though, over not memorizing that stuff.
No it wasn't, at least writ large in the USA.
And that's not what New Math was. Explaining what multiplication is doesn't demand an introduction to set theory.
If we read Feynman's CRITICISM of New Math, we actually find that he ADVOCATES for exactly what your parent is suggesting ("cobbled together.. in order to solve real-life problems"). So clearly, your parent isn't describing "New Math". Or perhaps Feynman is just a raving lunatic.
So I'm no advocate for "New Math", but I do oppose the argument you're making here, in which "New Math" is taken to mean "anything other than memorizing times tables" and is then denigrated on face. Without regard to the fact that the most vocal opponents of "New Math" were in fact advocating for exactly what your parent post is suggesting.
> Spectacular failure
So brief was the new math intervention that, to this day and despite all of the hoopla, we don't have a good empirical basis for claiming new math worked or did not work.
New Math was barely attempted, and its primary opponents were mathematically illiterate parents and teachers. This is just true, even if there were mathematically literate opponents to New Math, e.g., Kline or Feynman.
(But also note Meder’s reading of Kline. It's also worth noting that Mathematicians are maybe not the ultimate authority when discussing secondary pedagogy, especially in the mid 20th century. I have no basis for this belief, but IMO lots of mathematicians who weighed in on New Math were very possibly waging a sort of proxy battle as part of a larger war over the future of their own field -- pure vs applied.)
> Do you want to know what worked? Memorizing times tables.
Is this satire (honest question)? For all the things we don't know about math ed, we know that this doesn't work. Students who memorize times tables are routinely incapable of multiplying 12 by 13 or 55 by 55.
> There's just no way around the fact that drilling is key to early mathematical learning.
No, there isn't. But there's also no way around the fact that drilling without understanding is why a whole bunch of students who are "good at math" can't get through even the most dumbed-down versions of proof-based courses in college, or in some cases can't even get through a full calculus sequence. But they're "good at math" because they can rattle off 12*7 real fast!
New Math advocates (and their opponents!) were all at least correct about one thing: we REALLY SHOULD seriously ask what good is learning "math" if the student does not become a better problem solver. It's not 1417 anymore -- problem solving is important, but human calculators don't pull down living wages.
Probably the answer is that we should all be equal opportunity critics: memorization without understanding is intellectually lazy and limits growth potential, while understanding without practice is for most learners a contradiction in terms.
His beef with the New Math is with an emphasis on axioms, deductive reasoning, rigorous abstract logic, linguistic purity, and symbol manipulation, rather than with teaching conceptually or letting students think for themselves. He also doesn’t like the specific content of the New Math (set theory, inequalities, alternate number bases, boolean algebra, modular arithmetic). [I haven’t studied the New Math curriculum enough for myself to know how fair these arguments are.]
His key criticism: “Psychologically the teaching of abstractions first is all wrong. Indeed, a thorough understanding of the concrete must precede the abstract. Abstract concepts are meaningless unless one has many and diverse concrete interpretations well in mind. Premature abstractions fall on deaf ears.”
That obviously hasn't worked as most people are quite awful at math and society at large hates it. Memorizing times tables has been a spectacular failure as has memorizing formulas.
Math is about problem solving, not memorizing answers to common things; it's the focus on memorization that's made so many people bad at math to begin with. Common core is an attempt to address this by focusing on how the problem is solved rather than what the answer is, it's freaking parents out, but it is a better approach if you're actually trying to teach math.
So, here's the thing. Did I have no problems with math until uni because I'm smart? Because I have a specific type of brain? Because of my upbringing? Because I had very good education in my early years? Some combination of the above?
I struggle to believe that people can't do algebra. I am convinced that with some thought and help they'd find it trivial. However, is that my own experience overriding my ability to take their perspective as true? I dismiss "I can't" as "I don't think I can", but who am I to judge that better than them?
At the start of middle school, her teachers were recommending accelerated math classes, but she never ended up doing that, because she stopped liking math and stopped trying. Why? Maybe because it was unpopular or she didn't like her teacher... I don't know. In any case, she went from kinda-liking math to completely hating it over the couple of years, even though when I was there, she seemed to understand it just fine.
I really like him and I wanted to impress him with how smart I was and I remember putting in a lot of time doing all my homework. I got my only "A" in science for the first two quarters. Then he got fired and became a harley mechanic. Back the C's and D's
There are a lot of social factors in US schools, too. I notice a lot of young women doing poorly in algebra and precalc even though they're perfectly capable if you sit down with them and ask them to simply write down the steps of solving the problem. The social benefits of being "bad at math" can be seductive in the short terms and the long-term benefits of being good at math are entirely invisible to many young women. Guys at least see some male nerds on TV making good money; girls who want to be like nerdy tv stars know that they have to go into forensics and don't know that math is very handy for figuring out the rate of cooling of bodies and decay of tissue.
I used to have bad grades from classes I considered easy.
I hated math in middle-school and early high-school. It was boring and rote.
But proofs based math, and learning how to think of math as a language, instead of a collection of overly specific "solutions" crammed into my head by teachers, was the key.
It seems like an especially American thing - being good at math makes someone a freak and so it's something kids actively avoid at the ages in which being a freak is not tolerable.
That kind of attitude would need to be actively confronted because what happens is that kids who previously were developing skills tend to drop out given the peer-pressure.
Edit: best documenting link I could find but Google book won't let me copy a quote:
Our media praises stupidity
It's nice, vague, and populist. Everything thinks they know how to do critical thinking. Everyone thinks they know how it could be taught. I suspect most people think that they have above average critical thinking skills. As long as they don't have to agree on how or what critical thinking is, everyone can agree it's a good thing.
The other one that's dumb is "you learn best when you teach yourself". Sure, if you are really super-interested in something then you learn better, and tend to teach yourself a bit, but correlation is not causation.
I found their point fairly compelling, and I wonder if it applies to mathematics as well.
The episode was http://www.abc.net.au/radionational/programs/scienceshow/you...
It's more important to foster an positive learning environment and encourage/draw out curiosity and creativity at this age. Help them become curious about the world, fascinated with what they're learning and find enjoyment in learning.
In terms of subjects, aim for a strong reading focus at an early age as it will pay dividends in all other subjects (even math) if they can read/comprehend well.
I clearly recall memorizing the times tables when I was young. It suddenly dawned on me that "times" wasn't just a word, it meant adding the number so many times. I no longer needed to memorize the table, I just applied the rule.
Some university engineering classes present formulae and tell the students to just apply them. This is disastrous in my not-so-humble opinion. Where the formula comes from should always be taught, so students understand the formula and the assumptions it was derived from.
Recalling my own experiences with middle school and high school, I saw my peers frequently fall into the trap of trying to memorize pipelines of steps while we were learning algebra, geometry, trig, pre-calc, and then calculus. The ones who "didn't get it" would fall apart to varying degrees when confronted with problems where you had to think in terms of combining toolboxes of strategies to unfamiliar but analogous patterns, but I don't necessarily think it was their fault, because they didn't know better and rote memorization had worked consistently until this point.
I often feel a bit sad about it because I did a ton of proof based stuff outside of class (as a mathlete), and that was a HUGE part of my development in logical reasoning, intellectual rigor, and critical thinking.
Just anecdotal evidence, I know, but for me it was the exact contrary that happened. As a 5-year old kid I had learn to read all by myself (my dad had taught me to recognize the letters and how to pronounce them), but I had huge conceptual difficulties in teaching myself things like subtraction (what do you mean you can take/subtract a thing from another thing? what happens when the thing which you're taking/subtracting is bigger than the thing from which you're subtracting? you cannot do that, it would result in a non-thing, a thing that is below 0, and all things below 0 technically do not exist).
In elementary and middle-school I was passable in maths, while very good in humanities, but all this changed in high-school, mostly because of my teacher, who taught me (and my colleagues) how to think about math. He's one of the reasons why I decided to pursue a STEM education and why I'm currently a programmer.
To me the real important thing is the attitudes imparted by your parents and peers. As noted in the article, it's acceptable to fail at math. Parents who have their own history of math anxiety probably don't really push their kids to excel in that field. Just the notion that I was expected to "get it" and keep trying until I did (perhaps helped along by some natural stubbornness) may have made the difference in my case.
My conclusion is that every one will have strong and weak spots, and it's very important to stay alert to this and not beat around the bush thinking it's not important.
Took me more than a decade to get back into intuitive thinking, where you can play mentally with symbols and ideas just like LEGOs (as I was around high school).
This is the opposite of my experience, I did horribly in math and hated it until I took calculus (which, I guess I should have taken in highschool but I tried to avoid it since the rest of math sucked.)
EDIT: just want to clarify: I did have a reasonably intuitive understanding the whole time, the earlier stuff just felt like busy work.
My wife went to school for Architecture, where she learned "basic" structural mechanics, and some Calculus, but still cannot explain to me in simple words what an integral or a derivative is. Not her fault at all: her Calculus professor had them calculate polynomial derivatives for 3 months, without ever making them understand the concept of "rate or change", or what "infinitesimal" means.
For me that's a big failure of our current "science" education system: too much focus on stupid application of equations and formulas, and too little focus on actually comprehending the abstract concepts behind them.
I remember "hitting the wall" in Honors Algebra II, when I just couldn't keep up with all the new things I was expected to compute: radicals, synthetic division, logarithms, etc. It was a major dent in my math confidence, which up until that point had be reinforced by being told I was "good at math".
I didn't start regaining that confidence until Chemistry, when I learned what pH actually meant. That sparked the epiphany that scale was deeply important in nature, and with that motivation, logarithms seemed much simpler.
We so quickly forget that most math wasn't discovered as some abstract exercise. But we still teach it that way, with the concepts before the applications.
To top it all off, a lot of young college students taking precalculus are taking it as a terminal math class -- they'll never take calculus! It's just for a credit! So you've got some poor kid who hit that wall in high school and is repeating precalc and going through some unit on polar coordinates who will never take another math class again and will somewhat justifiably think this whole thing is stupid.
In theory, I strongly support "math for liberal arts"-type classes instead in college. When done well they might cover voting systems and how you can prove there isn't a perfect one, gerrymandering and how you can use math to get several solutions, perspective drawing and its relation to projective geometry and Renaissance art, a bit of number theory and basic cryptography....
I love it. They could call it "math"! ;)
Seriously, I think that captures an important concept: for the most part, math was developed to serve the needs of humanity.
I used to wonder, how do we know our numbers are the "true" numbers? Did we just happen upon one way of computing, and there are countless alternatives that could have been explored?
Of course, the answer to that is actually "kinda, yeah". Abstract algebra leads to all sorts of systems that behave more or less like our familiar numbers and operations. And besides the whole numbers, which represent physical quantities, every extension to things like integers, reals and complex numbers are used not because of their tangible reality, but because they do a good job of modeling human concerns. Because my brain has a limited ability to pursue abstraction for its own sake, its this aspect of math that really captivates me. I think that's true for most people.
I took a class that met in a room used by a class like that the period prior and got to listen to the better part of it. In the one lecture the prof coaxed them into creating an algorithm for the number of moves to solve a rubix cube based on the number of cells who's configuration is known and where those cells are located on the cube. In a different lecture they were doing the math necessary to cut hypoid gears on a manual milling machine. They didn't even know what they had did until he tied it back to the underlying concept.
The quality of those courses depends on the prof.
Entirely. In theory they are great :)
The one you listened in on sounds interesting!
potential (for attracting) Hydrogen. I had to look it up.
Funny enough, all the good ones were much younger professors (like, in their 30's, 40's), with very few exceptions. Since the original article is from the 80's, maybe Feynman's feedback had some effect? I did go to one of the best universities in the country, so maybe my experience is also different from other people's. I hope it isn't, and I hope it continues to get better over there.
A really good art teacher or a really good math teacher can really change the kind of future you'll have.
I've taught Calculus close to a dozen times now. I talked about integrals. I talked about derivatives. I tried to get into the intuitive nature of rates of change and the limiting process of Riemann sums. I used analogies and real-world examples and illustrations and interactive computer demonstrations.
Students don't listen to that part of the lecture. They just don't. They listen to the part where I start doing problems on the board, because to them math is about doing things, not concepts. That's only reinforced when they're asked on tests to just do things rather than demonstrate understanding of concepts. If, in a college class, you try to change that up by asking concept questions, you get bad end-of-semester evaluations because you've challenged them in a way they didn't expect and their parents aren't paying good money for them to be challenged.
(And realistically it's hard to ask concept questions at that level of mathematical maturity anyway. As answers you get mostly word salad with a smattering of misused jargon. I've tried.)
If you try to mix it up and use techniques from inquiry-based learning (Moore-style, flipped classroom, etc.) then you get a stern talking to from administration.
> , or what "infinitesimal" means.
Probably because even defining infinitesimals with anything resembling rigor is tricky, let alone using them productively. I wouldn't even try it in less than a junior or senior class. There is no such real number that is smaller (in absolute value) than every other real number yet bigger than zero. Infinitesimals, at the calculus level, are a convenient hand-wavy fiction to mask limits and a notational artifact from simpler times.
> For me that's a big failure of our current "science" education system: too much focus on stupid application of equations and formulas, and too little focus on actually comprehending the abstract concepts behind them.
I tried. Every single time I taught the class I tried to teach concepts. It's really easy to fall to the temptation of just working problems on the board every day, because trying anything else will just give you grief. Grief from the students, grief from your colleagues, grief from the administration. Grief from people like you who constantly chastise us for not being able to explain to teenagers that they should pay attention for the whole class. (I love my adult students though. Non-traditional students are the best.)
It's not worth rocking the educational boat if you don't already have tenure. If you do rock the boat you get a reputation for being hard or weird and students avoid your class anyway.
You hit a nerve. I've left academia for this and other reasons, but as you can tell I'm bitter about it.
Looks like I did! Sorry about that. I fully understand where you are coming from, as I've had the same experience when trying to help college colleagues with course work, tutoring for a few courses in college, or even giving private lessons to HS kids. Some people are 100% not interested in learning -- they want the bare minimum to pass the course and move on.
Not sure how we can change that, though.
I had one specific professor in college would would abuse those inquiry-based learning techniques that you have mentioned, and it was absolutely the best course I had during my 7 years in the university. Anecdotal, but students loved him. He ended up being my MSc supervisor :)
I disagree in the general sense. I think most of mathematics with which "normal people" (by that I mean people who don't get a masters or phd in mathematics) have problems is directly correlated to problems and situations in the real world. The problem is that we teach mathematics as if we were computers - which we are not. Its about formulas, recognizing patterns, precise answers and proofs instead of general understanding and intuitions about things - which is where humans excel and how we work and remember things.
For example, being able to solve quadratic equations is nice but IMHO what's much more important is knowing how the coefficients and their signs shape the plot of the lines and how that might effect the trajectory and hang-time of a projectile. Also being able to look at a projectile and give an approximate quadratic equation is again IMO more important than knowing all the steps in the quadratic equation.
But this is just my opinion.
I know the temptation is great to give lots of examples while teaching math, but you get the risk of only teaching little examples to students and not the abstract vision that math education demands.
Take your quadratic equations, it's all nice to demonstrate how coefficients shape the plot. But when you start talking about projectiles, you end up ignoring that quadratic equations are used... heavily... everywhere.
You may see them in chemistry, crypto or nuclear physics. But if your brain only remembers projectiles, it doesn't have the ability to abstract new problems the right way.
Now physics on the other hand, chemistry, computer science, any time I could apply math to a real-world situation got me motivated. I liked Math that could solve actual existent problems, and I learned said math better as a result. The first time I learned about projectile motion I literally started picturing vectors everywhere for the rest of the day.
It also didn't stop me from learning abstraction later in college. In basic signal analysis we learned about the Fourier transform, and then we learned about the Laplace transform and how it was simply a more general Fourier transform (or the reverse if you prefer). Mastering Fourier first gave the Laplace a relatable mental context, it made the abstractions more real.
By contrast I had a Math professor once who tried to teach us theoretical induction and work his way down to examples. I had no idea what the fuck he was talking about until he hit the examples, then everything clicked all the way back up.
Maybe that's just how I'm wired, but I would argue for more and more relevant examples. There are tons of historical uses of math that people would find interesting. Talk about the math used to build the Hoover dam, or telegraphs, fly satellites, or make medicines, or decode Nazi transmissions. And talk about it in real terms with character, if you must simplify for the sake of time then say that's what you're doing. Make people realize they're learning something that has uses beyond Math class or the rarified ivory towers of Math PhDs. That by learning the material they're learning how to do meaningful things. The original, wildly popular SimCity games were basically raw spreadsheets and formulae, but people learned those spreadsheets and formulae because the games had character.
I know to a lot of Math people the abstractions are the "real" part, but most people I know simply don't process the world that way.
I think the crux of the problem in teaching math is the need to illustrate each concept from MULTIPLE perspectives: theoretical, tangible, incremental, graphical, dynamic, etc. It's only by connecting the concepts to manifestations that most of us appreciate the meaning of math. Few are satisfied by disembodied equations or proofs. Most of us need some form of grounding using what we already know to make sense of new concepts. Otherwise math is just castles in the air.
Likewise for math. There is no "new book" to learn math that is going to blow the cover open on everything. It's a problem of motivating people to become fluent in math. I would like to see discussion in those terms -- fluency, because that's what it is.
This also applies to reading music and understanding music theory. Written music is a map of events/time - which is totally different than any other math I studied. I've noticed that a lot of musicians pick up programming pretty easily. I've often wondered if learning more symbolic systems that are different from natural language makes it easier to pick up additional systems like programming languages.
I love math because it's beautiful, primarily, and this isn't represented often in elementary or high school education. The idea of "function" or the idea of "constructive proof of bijection" are beautiful concepts. The patterns of math are beautiful (so beautiful I created an adult coloring book called Math with Crayons that focuses on Aztec diamonds, fully packed loops, lozenge tilings of hexagons known as totally symmetric self-complementary plane partitions -- you know, the stuff people are actually doing research on today!). Numbers to me are kind of boring. But how many kids get to draw fractals in class and then talk about the concept of dimension, and then argue about how one could rigorously define dimension, with the example of trying to define dimension for fractals as an edge case? How many kids get to draw non-crossing partitions and Dyck (lattice) paths and then try to prove that they are both enumerated by the Catalan numbers? How many kids get to think about the mathematical structures you allude to, and play with the fundamentals of category theory? Well, very few. And why?
Their teachers don't know any of that stuff.
If teachers only know computation, that's all they can teach.
If we had teachers who really had breadth and depth in math teaching math, I think we could start teaching concepts, structures, and beauty very early. I've had conversations about the basic ideas of category theory (relations not objects!) with many kinds of people. This is not out of the grasp of a kid. Set theory -- not out of the grasp of kids. The very idea of what a number is: in high school I got to take a seminar with other high school students on this, looking at John Stuart Mills and Frege and Russell and Whitehead and all this thinking about what a number is. But this is extraordinarily unusual material for a high school or even college student. Of course it's difficult to deal with these notions later, then -- people have been trained out of their natural curiosity and into a view of math as computation at worst and manipulation to find x at best, rather than seeing it as a way of dealing with beautiful structures.
Do you feel like you were taught how to think about math specifically, or how to think at all? Like probably most people here on HN, I love math and programming, but I'm struggling to recall whether and how I was taught to think in school. We went through a series of problems, and talked about the syntax and mechanics of solving those problems, using various frameworks (geometry, calculus, diff. eqs) So it feels like we learn math by example. Is learning how to think directly part of education, or only indirectly something that happens by seeing what others did?
Limits is incredibly easy to link to real physical things: movement, speed, acceleration.
If speed is distance over time, what's the speed of a moving object at a very specific instant?
Like I said, limits are used when a function "breaks", so they're naturally hard to visualize. I'll use my typical example for when I discuss teaching math by solving problems: Approximating the area under a curve using rectangles. The thinner your rectangles are, the more accurate your approximation is. So you narrow the rectangles, getting more and more accurate... Until you hit zero width, where the value of the function suddenly drops to zero... A discontinuity. A limit lets you find what the value would have been had that discontinuity not happened. So, there's a clear visualization of what the limit is letting you do in this particular case -- pretend like the rectangles of zero width don't have zero area -- but there's no clear generalization on how to visualize a limit.
"But, sir, I have only traveled twenty miles!"
I would rather see teachers accept that computers exist, and teach students how to solve problems, even very complicated ones, using software (e.g. Mathematica, R, or whatever). Keep the theory to an absolute minimum (leave the theory courses to people who are interested.)Really, so what if you don't learn how to do calculus manually?
It's about learning a set of thinking skills, not how to think. Many people who know no math can think and function very well in their domains and many people who know lots of math function and think poorly outside of math.
I'd go so far to disagree with you that people do well without learning math skill (those tools of thinking). If you can deconstruct a problem, then you certainly learned that skill somewhere, just not in a math class.
Abstraction and reductionism are unusual capabilities to acquire on your own. As I recall, they're almost nonexistent in pre-linguistic societies like hunter-gatherers. If you have no words for abstract concepts, your thinking will be strictly concrete.
- systematic (math, rule-based)
- social (history, emotions, people-based)
- creative (art, invention)
These all overlap and feed into each other because we're a complex species, but we really only ever hear about the necessity of math to "learn to think".
Source: I'm slow but good at Math and ended up dropping it as soon as I could because it would not get me the grades I needed to enter a top tier university.
I think that in a lot of disciplines, you have to become fluent at manipulations, and at seeing and thinking in higher level patterns. Being able to think your way through a more complex problem would seem to benefit from, if not require, seeing multiple steps ahead in a progression. At least this is my perception.
My experience in school math was that it wasn't enough to satisfy myself that I knew how to solve a problem. I then had to work my way through a whole bunch of similar problems until I could perform the manipulations quickly. This is also how I managed to commit the definitions, axioms, and theorems to memory. If I didn't do that stuff, then I got my arse handed to me on the exam. I gave my kids pretty much the same advice.
There's no shame in it (although high schoolers can be assholes), and it can help your school accommodate your needs eg extra time to take your exams.
My accuracy on the questions I got to was very high, I just couldn't go fast enough to complete enough questions. Same deal on SAT type math papers too.
Truthfully, my mention of accommodations isn't for you--I'm assuming you're out of high school already and you've found a professional niche that works for you. I mention test anxiety and the professional workarounds for high schoolers or their parents.
*) This is actually moving well beyond my expertise. I know about the existence of test anxiety, and I've accommodated students with it.
A brief consult with Dr. Google surfaced some clinically recommended accommodations that probably would have helped you. Sorry I don't have a time machine!
This one's my favorite: "Reducing the number of tasks required to demonstrate competence (such as 5 math problems instead of 25)"
From our experience most people struggle with math since they forgot/missed a curtain math skill they might have learned a year or two before. But most teaching methods only tell the students to practise more of the same.
When looking at good tutors, we could see that a tutor observes a student and then teaches them the missing skill before they actually go to the problem the student wanted help with. That seems to be a usefull/working approach.
Abstract reasoning, intuition, and creativity, to me, represent the underpinnings of software engineering, and really, most engineering and science, but are taught more by osmosis along side the unintuitive often boring mechanics of subjects. The difference between a good engineer of any sort and one that 'just knows the formulas' is the ability to fluently manipulate and reason with symbols and effects that don't necessarily have any relation or simple metaphor in the tangible world. And taking it further, creativity and intuition beyond dull calculation are the crucial art behind choosing the right hypothesis to investigate. Essentially, learning to 'see' in this non-spacial space of relations.
When I'm doing system engineering work, I don't think in terms of X Gb/s throughput and Y FLOPS... (until later at least) but in my mind I have a model of the information and data structures clicking and buzzing, like watching the gears of a clock, and I sort of visualize working with this, playing with changes. It wouldn't surprise me of most knowledge workers arrive have similar mental models of their own. But what I have observed is that people who have trouble with mathematics or coding aren't primed at all to 'see' abstractions in their minds eye. This skill takes years to cultivate, but, it seems that its cultivation is left entirely to chance by orthodox STEM education.
I was just thinking that this sort of thing could be approached a lot more deliberately and could yield very broad positive results in STEM teaching.
In other subjects you can rationalize to yourself in various ways: the teacher doesn't like me, or I got unlucky and they only asked the history questions I didn't know.
But with math, no rationalization is possible. There's no hope the teacher will go easy on you, or be happy that you got the gist of the solution.
Failure in math is often (but not always) a sign that education has failed in general. Teachers can be lazy or too nice and give good grades in art or history or reading to any student. But when the standardized math test comes around, there's no hiding from it (teacher or student).
People aren't afraid of making tiny mistakes. They are afraid of looking at a problem similar to a dozen they saw in lecture; and having no idea what to do, and then being told under no incertain terms that they did not succeed.
This may be grave heresy in the Temple of Tabula Rasa where most education policy is concocted, but nonetheless every teacher I ever knew was ultimately forced to chose between teaching real math class with a ~30% pass rate or a watered-down math Kabuki show with a pass rate just high enough to keep their admins' complaints to a low grumble.
In the end we teachers would all go about loudly professing to each other that "It's not about numbers, it's about learning how to think" in a desperate bid to quash our private suspicions that there's actually precious little that can be done to teach "how to think."
Start here http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf but then there is a vast literature (thousands of studies and other research papers) exploring the general topic of math pedagogy. Some approaches and some teachers are radically more successful than others, given comparable students.
If you hand me one of your 20th percentile students who is struggling but willing to learn, and give us the same amount of time which would otherwise be spent in a classroom for a year of one-on-one face-to-face time, with the student also spending a typical amount of time working independently, I can have them outperforming your 80th percentile students by the end of the year, without issue. The difficulty of course is that direct mentoring by an expert tutor is too expensive for society to be willing to pay for at scale.
The main reasons that your teacher friends are stuck is because (a) many if not most of their students are unprepared before they arrive in any particular course, which is largely down to structural social factors and school scheduling inflexibility, and (b) the combination of lectures and independent work with inadequate feedback are for the average student a terribly inefficient and ineffective way to learn. Neither of those has all too much to do with biological predestination or whatever.
However, I can well believe that a 95 IQ student who was dramatically unprepared at the beginning of the year and believes himself to be an irredeemable failure will have a lot of trouble in a standard lecture + homework drills format high school precalculus course taught by a teacher with no time to help him catch up on remedial material at his current level of understanding, give him any special attention, or make the class material engaging enough to give him reason to care. In that context, failure (or scraping by with an undeserved C) is the obvious outcome, but can’t fairly be blamed on the student’s effort during that year or some kind of mental deficiency.
(Personally, I skipped half of precalculus and somewhat wish I had skipped the other half; I found it to be an uninspired grab-bag of disconnected topics, with much too rote a focus.)
I was an Applied Math major at Berkely. Why?
When I was in 7th grade, I had an old school Russian math teacher. She was tough, not one for niceties, but extremely fair.
One day, being the typical smart ass that I was, I said, why the hell do I need to do this, I have 0 interest in Geometry.
Her answer completely changed my outlook and eventually was the reason why I took extensive math in HS and majored in math in college.
Instead of dismissing me, instead of just telling me to shut up and sit down, she explained things to me very calmly.
She said doing math beyond improving your math skills improves your reasoning ability. It's a workout for your brain and helps develop your logical thinking. Studying it now at a young age will help it become part of your intuition so that in the future you can reason about complex topics that require more than a moment's thoughts.
She really reached me on that day, took me a while to realize it. Wish I could have said thank you.
Wherever you are Ms. Zavesova, thank you.
Other beneits: doing hard math really builds up your tolerance for building hard problems. Reasoning through long problems, trying and failing, really requires a certain kind of stamina. My major definitely gave me this. I am a product manager now and while I don't code, I have an extremely easy time working with engineers to get stuff done.
It's impossible to tell if students are capable of thinking mathematically, however, because I have not met a single (non-mathlete) student who could give me the mathematical definition of... anything. How can we evaluate student's mathematical reasoning ability if they have zero mathematical objects about which to reason?
"You read all the time, right? We constantly have to read. If you're not someone who picks up a book, you have to read menus, you've got to read traffic signs, you've got to read instructions, you've got to read subtitles -- all sorts of things. But how often do you have to do any sort of complicated problem-solving with mathematics? The average person, not too often."
From this, two deductions:
• Having trouble remembering the quadratic equation formula doesn't mean you're not a "numbers-person."
• To remember your math skills, use them more often.
What I remember from high-school and college was this: I'd take a given math class (say, Algebra I) and learn it reasonably well. Then, summer vacation hits. Next term, taking Algebra II, all the Algebra I stuff is forgotten because, well, who uses Algebra I over their summer vacation? Now, Algebra II is harder than it should be because it builds on the previous stuff. Lather, rinse, repeat.
This is one reason I love Khan Academy so much. You can just pop over there anytime and spend a few minutes going back over stuff at any level, from basic freaking fractions, up through Calculus and Linear Algebra.
> But how often do you have to do any sort of complicated problem-solving with mathematics?
There are a LOT of chances to do that kind of complicated problem solving, especially if you're shopping or comparison shopping on your own. It's not that people don't have the chances, it's that people avoid the work involved in doing those kinds of comparisons.
Sure, if you go out of your way to actively look for reasons to find that stuff, you can find them. But my point is that when you're a kid learning math, it's summer vacation and you're busy playing with your friends, you aren't out actively looking for reasons to apply the quadratic formula, etc. At least not for most people, from what I've seen.
The article brings out a good point about math anxiety. I have had to deal with it a lot in my years of teaching math. Sometimes my classroom has seemed so full of math anxiety that you could cut it with a butter knife. I read one comment that advocated starting our children out even earlier on learning these skills, but the truth is the root of math anxiety in most people lies in being forced to try to learn it at too early an age. Most children's brains are not cognitively developed enough in the early grades to learn the concepts we are pushing at them, so when a child finds failure at being asked to do something he/she is not capable of doing, anxiety results and eventually becomes habit, a part of their basic self-concept and personality. What we should instead do is delay starting school until age 8 or even 9. Some people don't develop cognitively until 12. Sweden recently raised their mandatory school age to 7 because of what the research has been telling us about this.
In middle school in the US you start getting teachers specialized in their teaching subject matter, but often tangentially. Any one with enough college math credits can teach middle school math (with some other certifications, usually). High school you may, finally, get a math teacher who is a mathematician. If you're lucky.
People ask me if they see the beauty in math when I'm complaining. I do, but I also see the other kids outside playing while I'm held back from recess until I can tell you what 7*8 equals. Fuck that so much.
Interestingly, now as a masters student in a statistics graduate program, I've learned that I don't like "doing" math but get enjoyment from teaching it. I really like it when students challenge me when I'm at the chalkboard and I'll do anything for those "ah-ha!" moments. The best is at the end of the semester hearing students say "I thought this class was going to suck but I worked hard and am proud of the work I did." I'm hoping that on some small scale I'm shaping their views on math. Or at least give them the confidence to say, "I don't get this, but I'm not afraid to learn it."
By that I don't mean it's easy. But when you're grappling with some problem, whatever it is, eg find some angle or integrate some function, if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?"
And you won't have any excuses for why you didn't think of it. Because math is a bunch of little logical steps. If you'd followed them, you'd have gotten everything right.
Which is a good reason to feel stupid.
But don't worry. There are things that mathematicians, real ones with PhDs, will discover in the future. By taking a number of little logical steps that haven't been taken yet. They could have gone that way towards the next big theorem, but they haven't done it yet for whatever reason (eg there's a LOT of connections to be made).
"The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles."
Really? There are many times where I thought "How am I supposed to think of that?" In fact, that's probably what would happen most of the time.
The problem with high school math is that it has no context. It's just a bunch of seemingly arbitrary rules.
It's as if when learning programming, you would learn fragments of various languages here and there but never connect the various pieces into anything meaningful.
Lesson 1: boolean variables in Java
Lesson 2: If statements in python
Lesson 3: Three-part for-loop in C
Q1: In the C language, how do you separate the components of a for loop? (correct answer: semicolons)
Q2: What can you put in the right hand side of the '=' symbol in a boolean variable declaration in Java? (correct answer: an expression that evaluates to either true or false).
Q3: You are supplied a function that can take any number and determine if it's a Crastomarian number. We have 20 foot ball players each wearing a T-shirt with its own number. The numbers go from 5 to 24. We want to find how many players have Crastomarian numbers. The program to find the correct answer will use a for loop. Write the header part of the for loop. (correct answer: `for(i = 5; i < 25; i++)`)
And this is supposed to be how you learn programming.
Also note: there's almost no way to know the answers without rote memorization. Of course you will never absorb these concept by osmosis because your day to day activity will not involve writing programs that do things. Instead, you're just memorizing the rules given to you and doing exercises that help you recall which rule to use for which situation.
If you haven't memorized the for syntax and written 25 different variations of it, you will not be able to know it.
And when I say that you write 25 different variations, I mean you write them as part of pointless exercises of course because these are not in the context of a program.
It's more like just repeating these lines until the pattern somehow hopefully sinks into your mind.
for(x = y; x < z; x++)
for(y = 10; y < j; j--, y++)
My University computer science was also broken up as you describe. The introductory class was in Pascal, the next course was Java, and then each course used its own language.
That's not what I'm talking about at all.
Too many students lack the ability to perform basic algebra. I mean simple stuff like simplification, factoring, working with exponents and even fractions. Faced with a typical problem in calculus, of course they're going to flounder and make errors when each calculus problem involves perhaps a dozen simple algebraic manipulations.
These basic things have to be drilled and practiced A LOT before moving on to more advanced topics. There is no way around that. No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.
Conversely, no amount of memorization and drilling will provide students with the intuition necessary to solve problems just beyond the grasp of naive symbolically-driven proof search algorithms.
In a typical non-proof-based calculus sequence, tricky u substitution and integration by parts problems are the two places where you really see "I was good at math in high school" students start to struggle because they lack a good conceptual foundation in trig and logarithms. Up until that point the students with trig identities etched into their skulls by practice and drilling do just fine. But from that point onward, they really start to struggle because they need a bit of geometric intuition and some practice with more effective problem solving techniques than "blindly apply equations until something works".
Really, both are needed -- drilling and practicing, as well as a solid (and preferably proof-backed) intuition about elementary properties of trig funcations and logarithms.
With the sole exception of Geometry, every single math class I took in middle and high school was an absolutely miserable time of rote memorization and soul-crushing "do this same problem 100 times" busy work. Geometry, meanwhile, taught me about proofs and theorems v. postulates and actually using logical reasoning. Unsurprisingly, Geometry was the one and only math class I ever actually enjoyed.
All arithmetic is math, but not all math is arithmetic.
Except that if you dig deep enough, all math is arithmetic under the hood. Algebra is doing the arithmetic on variables before they're numbers. Calculus is just useful shorthand for compound arithmetic; the derivative is a divide and the integral is a sum. Exponentiation is repeated multiplication, which is repeated addition. Linear algebra is arithmetic on blocks of numbers.
So we abstract over the individual arithmetic operations for our own convenience, there are too many of them. But it's all still fancy arithmetic.
Maybe we should call math "fancy arithmetic" to help all the people who are intimidated by math...
Seems like you agree more than disagree; math and arithmetic aren't easily separable, if you're being serious instead of cheeky?
1) Arithmetic (number theory) is far harder than most people realise, to the point that its considered one of the most complicated mathematical disciplines.
2) Most subject areas in math bear little to no resemblance to arithmetic, so it doesn't make sense to call all math 'Fancy Arithmetic'.
But I'm not actually proposing that, just sharing a cheeky point of view to contrast @taneq's boss, so don't take me too seriously. ;)
I am a newb at this stuff, but I believe (basic) number theory actually builds on recursive structures, and then uses induction to make statements about behavior.
What is interesting is how the same approach works for analogous structures like lists - because of the monoid abstraction.
But that said, I didn't learn addition in terms of monoids, and I don't know anyone else who did. This might be a grand unified theory of math under which addition fits, but physical addition has no recursive monoid analogue. You can add two weights together to get a measurable sum, and it does not depend on a recursive structure that uses induction. You just put two separate things side by side on the scale.
Just because we can explain addition using monoids and induction does not mean that addition is made of monoids and induction.
Also, I probably wouldn't use Peano theory as the starting place to teach addition. As a pedagogical tool, this is advanced math, and I guess would be a bigger impediment to learning than Greek symbols, for beginners. Right?
I mean, there is theory & practice in every field, technique and discovery. A medical scientist studying the development of aberrant cells in a petry dish has a pretty different job to a pediatrician or radiologist. They generally seem to coexist happily under the "medicine" label. For first year undergraduates, they would probably need to take most of the same classes.
I love talking about both subjects I went to university for actually and both can be difficult to get people interested in. There are massive misconceptions about both.
- They see a shopping bill or a phone number and ask you if you can add them.
- They see random equations, often in the background, and ask you if you understand what it means.
- arithmetics (a.k.a. Number Theory) is a vast branch of modern pure mathematics;
- numeric calculations, i.e. how to do them efficiently, is a very important part of the (applied) mathematics.
The ONLY sane way to answer these questions:
- Does math increase critical thinking?
- Does critical thinking lead to more career earnings/happiness/etc?
- When does math education increase critical thinking most?
- What kind of math education increases critical thinking?
Is with a large-scale research study that defines an objective way to measure critical thinking and controls for relevant variables.
Meaning you don't get an anecdotal opinion on the matter on your study-of-1 no-control-group no-objective-measure personal experience.
I think I often struggled or was intimidated by the syntax of math. I started web development after years of thinking I just wasn't a math person. When looking at this repo, I was surprised at how much more easily and naturally I was able to grasp concepts in code compared to being introduced to them in math classes.
That series of book really put intuition at the forefront. I began to realize that the crazy symbols and formulas were stand-in for living, breathing dynamic systems: number transformers. Each formula and symbol represented an action. Once I understood Math as a way to encode useful number transformation, it all clicked. Those rules and functions were encoded after a person came up with something they wanted to do. The formula or function is merely a compact way of describing this dynamic system to other people.
The irony was I always thought math was boring. In retrospect it was because it was taught as if it had no purpose other than to provide useless mental exercise. Once I started realizing that derivatives are used all around me to do cool shit, I was inspired to learn how they worked because I wanted to use them to do cool shit too. I went through several years of math courses and none of them even attempted to tell me that math was just a way to represent cool real world things. It took a $10 used book from amazon to do that. Ain't life grand?
> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.
In mathematics, intuition is the starting point and guiding principle, but never worth anything by itself without rigour.