Meet the New Math, Unlike the Old Math
quantamagazine.org
quantamagazine.org
My 11 and 13yr olds are dealing with this Common Core bullshit right now. This article might give an insight into why CC is so bad, yet they think its so good. Like when we demo a software product that we've all been working on for six months, and discover that users can't figure out how to make it work: it only works for us because we've been using it continuously for six months. Maybe Common Core can teach Proportionality if you sit down with a kid and give them all six years of it over two days. What happens in reality, is that my kids (a) don't understand the confusing ways its presented and (b) don't make the connection between the different expressions of the same underlying concept. Its not useful. Its actually immensely frustrating for everyone.
I think they have it exactly backwards. Teach the algorithm. Rote practice. Then teach the concept.
I'm reminded of my first year thermodynamics lecture at Imperial: "We're going to teach this to you every year. The first year you wont understand it. The second year you'll think you understand it. The final year you'll realize you never really understood it, but at least you'll be able to use it."
And yes, the rote practice is what makes it sink in. I just also need the WHY so that it makes 'sense' what I'm doing and to help me understand why I am doing it.
https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
I was simultaneously thrilled that these concepts actually had meaning and annoyed that no one explained this meaning to me at the time. Math is filled with definitions and proofs, but none of that has any point unless the result actually corresponds to something interesting!
(btw I don't have any experience of learning from common core, so I don't know if I would like it or not)
By the way, I have a friend who's a TA in statistics and this is a big point of frustration for them. Undergrads are starting to do just what you said : unquestioningly follow intuition. It's a disaster. Not that there wasn't always 50% of the class doing this, it's just that it's now 90%.
Many classes are still stuck in teaching students to pattern match rather than really understand what they are doing, and why. Practice on a bunch problem sets without does nothing to fix that.
Rote learning is just not useful for the jobs that are humans are going to do next, where computers are going to be able to pattern match much better than us anyways.
My theory is that this version of Math is designed by people who hate Math (English teachers). It's all about estimation and fuzzy thinking. Everything that I personally liked about Math as a kid seems to have been stripped away. In my opinion, children who want to become computer programmers, scientists, and engineers are probably not well serviced by this type of teaching.
I'm not saying the old method was perfect either; mostly because if you're just teaching algorithms then there isn't much understanding about why they actually work.
The issue is that machine learning, at least in the modelling domain, is exceeding the maximum complexity the human mind can contain. Attempting to analyze neural networks is fun, and it's what you do as a researcher. But no amount of study is going to tell you what more than 10 neurons are doing to any reasonable amount of detail, no matter how smart you are. Machine learning algorithms suck at explaining what they're doing mostly because what they're doing is far more complex than what we do. And yes, I can build toy examples like a 100 bit binary adder where I do know, I'm talking about non-obvious cases.
Therefore: rote is all that matters. You should be able to execute these things, forwards and backwards and take it's code apart and put it back together again. That's all there is to it. And if machine learning does indeed conquer the world, that's all there will be to anything.
Also, unless our current machine learning algorithms just need bigger computers to take over the world, somebody is going to have to develop more. If some kind of guiding intuition isn't a part of that process, the universe will hit heat death before we build an AGI.
You might think that, but you'd be wrong. "Precision" is explicitly one the 8 principles of Common Core Math.
https://en.wikipedia.org/wiki/Common_Core_State_Standards_In...
> In my opinion, children who want to become computer programmers, scientists, and engineers are probably not well serviced by this type of teaching.
Do you have any data or evidence regarding this opinion, or is it just your "estimation and fuzzy thinking"?
master: follows the rules because he understands them
guru: transcends the rules because he understands where they don't apply
http://nautil.us/issue/40/learning/how-i-rewired-my-brain-to...
The stigma against rote practice has damaged my past learning efforts. I thought I was learning things the wrong way. :-(
The efforts to de-emphasize it seem to be in the hope that it'll magically cause people to learn the higher concepts as long as you use the rest of the time to read passages out of a book about them. They've clearly been unsuccessful.
I agree. I remember how I learned: lots rote exercises and word problems. Clearly recall having insights and making discoveries on my own, but those made sense based on how I visualized things. And because of discovering them on my own, they were very memorable and fun.
The funny thing, repetition is how kids learn. They copy and repeat things since day 0 without understanding why or how anything works. Insights and understanding the meaning of things come later. Reversing that process is counter-productive and goes against all the evolutionary mechanisms optimized up to this point. That kind of build-up from first principle will be useful, but that's closer to highschool or college level.
It sounds like you're poo-pooing the centrality and depth of notions of proportionality in Mathematics.
Just in case that was your intention: expecting a student's understanding of proportionality to fully mature over 6 years is ridiculously optimistic -- often a big chunk of the first year of a graduate mathematics program is devoted to introducing measure theory.
> Maybe Common Core can teach Proportionality if you sit down with a kid and give them all six years of it over two days
Hell no it can't. How in god's name are you going to teach about fractions and the Riemann integral in two days?
To be clear, proportionality is extremely important in both cases -- I'm not sure what it would even mean to successfully teach either of those things without your students noticing something essential about proportionality.
Obviously, a competent teacher will notice that some analogies and connections could be useful to point out and elucidate, while some are still a decade away from being useful to the student.
> Teach the algorithm. Rote practice. Then teach the concept.
This isn't at all contradictory with Common Core... what makes you think it is?
For me, "State and motivate the concept. Teach the algorithm. Rote practice. Then teach the concept. Rote practice with the concept as opposed to the algorithm." is far more effective, but anyways, the practice and algorithm is still there.
> What happens in reality, is that my kids (a) don't understand the confusing ways its presented
Oh right, shitty execution. I'd be unsurprised if the teacher is as confused as everyone else.
You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME.
I know why I'm not a teacher...
> and (b) don't make the connection between the different expressions of the same underlying concept... Its actually immensely frustrating for everyone.
Well, yeah, learning hard stuff usually is.
First off, I'm not getting the proportionality => measure theory link. Proportionality figures in much more prominently in other areas (e.g. projective spaces, geometry in general). Where's the time you allot for studying PU(n)?
Second, you forgot to demand that high school science teach kids how to derive hyperfine structure so they can really understand what a hydrogen atom is. You should probably give up on teaching them about any of the other elements at all.
> You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME.
Except what you'll get is as many opinions on what to spend the 6 years on as there are mathematicians. And any discussion is likely to be dominated by group theorists thinking group theory is the most important thing mankind has come up with, analysts thinking the same about calculus, etc. Doesn't sound helpful to me at all.
1. Some ideas (eg proportionality) recur throughout a course of study in mathematics. Pointing out the relationships between these ideas is absolutely a useful thing to do, and a student's understanding of these "simple" concepts should mature throughout their study.
That's not some profound debatable point. It's just a really stupid simple observation. CC is full of these stupid simple observations that somehow become controversial because of shitty execution.
2. Shitty execution of CC is an indict of our inability to make education an attractive career choice, far more than it is an indict of CC. Having students memorize relationships between angles and points on a unit circle without making sure they can derive those facts themselves is exactly as useless as showing the derivation without giving them any practice at calculating things themselves, for example.
But at least CC gets one thing right -- ideas matter. And yes, you also need practice. But rolling back CC (or adopting it) isn't going to solve the fundamental problem with math education in the US, which is that most people who are really good at math don't become math teachers. For obvious reasons.
> I'm not getting the proportionality => measure theory link
Do you get the link between proportionality and Lebesgue integration? Then the connection back to measure should be obvious. (edit: better example)
My point is that proportionality isn't some "stupid simple" concept that CC has somehow over-complicated. It actually does recur as an idea of central importance throughout mathematics, and drawing out those connections is a good use of instructional time.
> Second, you forgot to demand...
That's a stupid demand. Teaching measure theory in high school as part of a standard curriculum is also stupid.
But that doesn't mean that idea of proportionality don't recur, at varying levels of sophistication, throughout a standard age-appropriate course of study.
> Except what you'll get is as many opinions on what to spend the 6 years on
Right, well, that's kind of the whole point of having a standardized curriculum...
Also, empirically, I've literally never seen this happen. Only in the context of after-school or extra-curricular enrichment stuff do people focus on topics they're passionate about, and in those cases it's entirely appropriate.
I think this is currently the main obstacle with the CCSSM and NGSS. The structure and logic of the news standards are good, but they're new so it will take some time for teachers to adjust. We need better course plans, better textbooks for teachers, better textbooks for students, and better exercise frameworks.
We can't look to mainstream publishers to provide this, because they're large, slow, and extractive organizations. Therefore, this seismic shift in the structure and material taught represents a tremendous opportunity for OER to come in and eat the big publishers lunch, e.g. [1].
Our company (Minireference Co.) is currently focussed on university-level textbooks so we can't "attack" the high-school market, but we're happy to share technology and insights with anyone out there who is working on OER for primary/middle/high school.
[1] State-sponsored content push to create common-core-aligned textbooks for high schools: http://k12oercollaborative.org/
It's actually much worse than this, the only math teachers that actually understand how to teach this material are math teacher teachers and their newly graduated students who have gone through their curriculum. It's going to take a literal lifetime to actually meaningfully execute the new methods even if you manage to get the textbooks in every classroom.
I really truly hope the kinds of resources you're developing can help existing teachers transition, because as it currently stands the outlook is bleak.
"a hideous atrocity that has been proven to hurt young children's minds" is extremely alarmist. I don't think that helps conversation, but I can understand your passion.
Where have you seen that Common Core is harmful? Most sources in my (admittedly limited) research suggest that it is far to early to tell an effect.
The most cited report seems to be https://www.brookings.edu/research/a-progress-report-on-the-... . However, the data reported here is extremely messy. Looking at table 3-2 the conclusion that CC may be harmful is drawn from the fact that n=1 has an improvement of ~2.25, whereas n=5 has an improvement of only 1.94. However, n=4 is at -0.83, n=3 at 1.8 etc. Results are all over the map.
In my opinion there is no clear pattern, let alone a "proven" result. If there is research I have missed I would genuinely like to hear it. I don't have a fixed opinion.
Then, the "new math" was invented, and people concluded education was broken.
Now, "common core" and it's deja vu all over again.
I don't have high hopes for education reform movements, either. The direction education needs to go to get better is simply "less", and there's no incentives for doing so. (After all, how can you get credit for "helping children" if you're not actively doing anything?).
This is probably exactly what administrators want, though. Cheap teachers, and damn the consequences.
Are you kidding? Administrators live or die on test scores, and the non-brain-dead ones realize that good teachers are the best way to get those test scores.
They don't really have a choice, though -- they work with what they're given in tax money and in many states don't even have any autonomy in compensation (NEA).
IMO the problem is way earlier in the pipeline, anyways. Even if administrators deeply cared about quality, they'd find a shallow pool. As a good rule of thumb, people who are good at mathematics don't become teachers. They become engineers or go into finance and consulting or write software.
And it's all about pay and quality of work environment. I'd teach high school math in a heart beat if the working conditions were more reasonable and it paid even half as well as my other options.
IMO category theory would be a waste of time without first going through a mathematics undergraduate course of study.
Getting past the preface of a reasonable book on category theory requires internalizing the idea of a natural transformation. If you can't get that far, you won't get anything non-superficial out of studying category theory.
Understanding what a natural transformation is requires at least one example you can relate to. Understanding why it'd be useful to abstract away from that particular example requires knowing enough about at least one other example.
And to do anything even moderately interesting in category theory you need a fair bit of mathematical maturity, which means being able to write and read proofs together with a bit of knowledge about modern logic, analysis and algebra. Without that, a lot of the motivation for category theory is kind of lost (and probably most of the available exercise aren't even comprehensible. And learning math without exercises is close to impossible.).
You'd have to pick your topics carefully and probably reformulate a lot of definitions. And you'd need the student to think it was cool for its own sake. But I could see this working out.
The song is occasionally used as an argument for going back to the old way, but it seems somewhat contradictory when parents of school-aged children aren't old enough to have experienced the old style of math instruction he advocated for. New math was a trend from the 1960s, so you'd have to be pushing 70 to have experienced elementary school math before it.
Of course, this sort of tension is present in a lot of academics. Focus too much on the theory and students can't actually do anything useful; and many will be completely lost. Focus too much on the rote learning and the students don't actually understand any underlying principles (and, in the case of arithmetic, can't actually do anything a $5 calculator can't today).
Sometimes people question the value of teaching math to kids. But I can say anecdotally that the people I've met who did not understand arithmetic correlated strongly with people who were disasters with their personal financial affairs.
(And it never did me any harm. Twitch, twitch.)
I learned different bases, sets, and the things I associate with New Math in either the limited high school computer classes or college. To be honest, I don't know if that's good or bad.
[1] http://www.nationalreview.com/article/373840/ten-dumbest-com...
In other words, the Common Core seems to get flak because some of the material is simply bad and sometimes seems not to have good idea how to convey the Common Core ideas.