Math = Numeracy, logic, geometry. Those are the three reasons I learned math. All three benefit from memorization. All three are practical. All three speed up future thinking. All three are scalable. All three are intuitive. All three are thinks you can't live without.
The reason why math is 'hard' is the same reason 'exercise' is hard. It takes a combination of skill and practice (a/k/a discipline) to make it 'easy'. And in a similar way to the endless "how to lose weight" fads, we see "how to do make math easy," when <easy> is not 'in their nature'.
The main problem with math education is that it is falsifiable. Math teachers can be measured, because math performance can be measured. This is why it makes educators nervous. And, I suspect, why they mostly suck at teaching it. They are the ultimate risk-averse sub-population. Subconsciously, 'educators' <hate> the idea of accountability, because it is predicated on the notion of potential failure. But the 'educational' establishement is <built> on the foundation of unquestionable <authority>. so math is for these guys an existential threat.
TLDR: To be good at teaching math, you have to be open to measuring your failure, which makes people feel bad. Including the teachers.
No. No no no. Yes, arithmetic math does benefit from repetition and memorization, is useful, and your comments on why getting good at arithmetic is hard are spot on.
But math is so much more than arithmetic. Math is about learning to apply a variety of available solutions and processes to solve a problem. Math is about understanding relationships. It's why I get better at math in a combination of my physics and programming courses than I ever did in algebra or calculus.
In fact, the biggest problem with mathematics education is that it is not falsifiable. Measuring reasoning, logic, critical thinking, and problem solving skills is almost as hard as teaching them. Writing a generally applicable test that measures whether a teacher taught you to actually use and solve problems with math is nigh-impossible, made worse that students (and most of all, parents) don't want to be held accountable for problem solving, because it's too fuzzy, or because you just "have to be smart". The fact that real math skills are tough to teach and tough to measure is the problem, not the reverse.
Math is much more than arithmetic as it is taught. Understanding arithmetic through the lens of relationships (as you stated) is no different than algebra, geometry, trigonometry, calculus, differential equations...
Arithmetic is important, especially in applications. But I wouldn't say it forms the backbone of all mathematics. You can do symbolic calculations just fine without any actual numbers. And a number of fields don't even involve any numbers at all, not even in applications.
Another problem with math education is that many of the people in leadership positions in public school systems are, when it come to math and related skills, functionally illiterate. Those people see themselves as successful and are respected as leaders but have little use for math themselves. It's hard, then, for them to believe that an understanding of math is genuinely essential and worth fighting for.
Your exercise analogy is perfect. I think your concept of Math is in direct opposition to the author's (and mine).
The author is arguing that math is about logically solving problems based on a small set of simple rules. That students memorizing a formula are learning a formula but not learning Math. Like you said, Math is like exercise, where you challenge yourself with harder problems and over time you become a better problem solver. The shortcoming with American public schools, or whatever we're talking about here, is that the teacher's incentive is to get their student to achieve a competency in repeating memorized instructions, something that can and is tested. This leads to numerous students who don't enjoy or understand Math for what it really is, and fail to develop the part of their brain that solves problems logically. Instead, they learn a bunch of formulas that are totally pointless unless they happen to need that exact formula in the future. So, evaluating teachers based on the performance of their students at applying different formulas is part of the problem, not the solution.
I'm not sure we disagree too much. My view is that you need to teach strong fundamentals so you don't need to think (ie, worry) about them later. This reduces degrees of freedom, and frees up your mind. The reason you do this is precisely that it lets you focus on higher level abstractions. It might just be me (probably is) but I found that math got more intuitive this way. Also, if your aim is to make the basic building blocks both reliable and intuitive, it will get used more in daily life. If you need to reference a text, it will ~never happen. [1,2]
____________
[1] What are <legos>, but simple geometric objects?
[2] Life is an open book test, there is just not time or the bandwidth. And I'm not talking about memorizing a calculus text, etc just for fun. But more basic - the bulding blocks. But if you want to play for high stakes later on this is the Ante, IMHO. That probaly wasn't clear in the initial comment I made (as several people have pointed out logic~abstraction, etc). So I'm not saying this should be the limit or ceiling of an education. Its just the opposite. But its shocking the kinds of things that happen otherwise. And its not fair to the kids who never get a shot at doing serious work, because their fundamentals are unsound. "Subtle bigotry of low expectations" and all that.
It's so unmathematical to use the equal sign there... and of course there are plenty of people who live their entire lives without any of those three things.
Math is also about finding structure and patterns that don't arise in any of those three contexts. It's about learning how to generalize and specify. It can be about isolating the relevant parts of a real-world problem so you can solve it in a way that matters.
None of those skills involve or benefit from memorization, but they're damn well more important than memorizing a multiplication table or knowing facts about convex polygons.
There is a lot of value in this comment. For example, the notion of abstraction as seperate from logic. Isolation and prioritization. etc. The importance of basic structures.
But: I'd be carful with language.
Example (1)
None of those skills involve or benefit from memorization
But consider <Memory> management is important from the perspective of <cognitive science>. Memorization helps you process and prioritize. Consider the notion of abstracting structure from data, using pattern matching. How does one get better at this? (1) you have empirical data; or (2) you process it efficiently. Memorization facilitates data processing. It also provides for important empirical data-sets and a means of prioritization both of which are crucial for heuristic reasoning. Not to mention formal modes of analysis.
Example (2)
Math is also about finding structure and patterns that don't arise in any of those three contexts.
Where do you think the "pattern" in pattern matching comes from (if not Geometry)? It's the basic things that are most often overlooked. Consider the most elementary use of applied statistics: Linear regression. The abstraction Y=mX+B? That has nothing to do with geometry, rigt? The point is that these most basic things are <critical> to more advanced concepts. Things you seem to take for granted, but are things of great value to the novice and advanced practitioner alike. And if they are not learned (so they can be "forgotten" in memory) they are <impediments> to progress for many students.
If you are innumerrate, you're going to have trouble with your bills and interest rates and such; if you can't apply logic then you'll have trouble holding a job where you have to make a lot of decisions; but geometry? How many people are concerned with geometry beyond the most rudimentary concepts like interpreting a street map. I just think statements like 'you can't live without geometry' further the 'disconnect' (as they say in America) between the people who 'get' maths and the people who don't. It's almost like a religious person telling you that 'you need Jesus in your life' or something - it doesn't actually persuade in any way, just signals that this person maintains some sort of alien, abstract belief system which they cannot relate to the lives of ordinary people.
Also, Physics: The arch. A triangular truss. A lever.
Also, Architecture: Lines, planes, volumes.
These are some very basic (perhaps 'unremarkable') everyday uses of geometry. I don't think at all the fact that they are rudimentary is a problem. Because they are so pervasive, it is all the more important to have mastery of the subject. IMHO.
With that example we should be teaching our high school students graph theory. And I'm completely in support of that. Nobody uses geometry to read a map, because geometry is about shapes and angles, not routes.
Students need a motivating context, a reason to learn mathematics (or any topic). It's called situated learning (also situated cognition, situated action) - with techniques like problem-based learning, challenge-based learning, service learning, project-based learning, learning by design, learning through games, simulations, modeling, programming, etc.
OK, I know a bit about pedagogy, and a lot of it is crap. There's an awful lot of relativism in most of it. Many of the people who teach teachers simply don't care about what works in classrooms, because empiricism is an evil capitalist plot (or something like that). When the Wikipedia page for Education has gems like "Based on the works of Jung", you know you're not in a field which cares about hard facts. Not that it's a real problem - the statistics show that first year teachers are pretty poor (regardless of the amount of pedagogy training they have), but they learn on the job and are pretty good after about 2 years (unless they are just doing a year of teaching to pad their resume, but that's another debate).
I've got a lot of respect for cognitive science though, and that seems to be what you are recommending.
Note that Jung was one of the first people to invetigate the use of words in schizophrenia (in his thesis, as a matter of fact). Secondly, he created the conception of extroverts and introverts that we typically use to describe people today.
Seriously, go read some of the works of Jung before you sound off on his issues. I did, and I was pleasantly surprised at just how sensible he was.
I completely agree that much of modern educational training is quite poor, but take aim at (for example, the Myers Briggs or visual, auditory, kinesthetic learning) rather than poor old Jung.
Except the examples you've given like extroverts vs introverts are not scientifically rigorous and the terms already existed well before him all around the world in different cultures (outgoing vs shy/retiring etc).
I consider science to be the art of making correct predictions. So you need a hypothesis (in the form of equations or computer programs) that can be tested on observations leading to correct predictions.
Extroversion/introversion might be scientifically rigorours by that definition (they might have done quantitative studies) but I'm not sure Jung did those, and even that it makes sense to create such a classificaton - people are a lot more complex and I don't really see the need for use of such classifications.
That's a problem I see with psychology, sociology, economics, and other "soft" fields. The quantification of things which are far more complex than the simplistic models created to the point that they are essentially meaningless. It's meaningless to quantify people into races or skin colors and that's about the level that those sciences are at.
Jung did his thesis in 1903. In 1903, regression had been invented, but was in England and may not have spread very far from there. Fisher was 13, and had not yet invented maximum likelihood and the application of statistics to experimental design that we all benefit from today. Karl Pearson was working on statistics, and had invented the chi squared three years previously. He was in the process of generalising regression analysis in 1903.
Multiple regression (OLS) had not yet been invented. I'm not sure how you expect Jung to have used an apparatus of statistics which really wasn't developed (till Fisher) into a coherent whole when Jung would have been in his forties, having already written a number of books.
Statistics in the sense of looking at populations did exist, but the whole apparatus of modern statistics was developed in the early part of the 20th century, concurrent with Jung.
I agree with your definition of science. Incidnetally, Hans Eysenck developed a physiological test for extraversion/ introversion in the 1950's involving the amount of stimulus required to become noticeable to a person. This has been comfirmed by further research.
Speaking as a (soon to be) psychology PhD (all going well...) I would argue that the "soft" sciences have the exact opposite problem, in that, jealous of all the cool theories of the physicists they have attempted to jump straight to the theory building without the benefits of hundreds of years of observation.
I agree that models in the soft sciences are somewhat simplistic, but I actually think that they're not simplistic enough. We (as a species) need to figure out some invariants if we're ever going to do successful science on people and the systems we create.
Back to Jung, while he didnt use the statistics that we would today, he did spend an awful lot of time attempting to figure out why we are the way we are, without resorting to sex sex sex (like the inimitable Freud). Personally, at this point he's probably better read as a philosopher and student of human nature, but he is well worth reading in that capacity.
That being said, he's an awful writer so it is a bit of struggle. Well worth it though, in my opinion.
Replying to myself as I believe we have triggered the algorithm for shouting matches (though I don't believe either of us were shouting). Dangers of fully automated approaches (black boxes), I suppose.
Funnily enough, I agree with you on psychologists and psychiatrists (to a certain extent). We understand so little, and claim to know so much. Our sample sizes and cultural range is quite poor (anthropologists are good at this, but they tend to lack even a basic understanding of statistics). I do believe that algorithmic approaches to predicting humans have potential, and the reason I now work in the private sector is to get access to some of this data as I believe that masses of data are the only way we'll get invariants to form useful landmarks towards understanding of people.
I would note, however, that I suspect you are classing pychologists (or cognitive scientists, as some of the hip american departments have rebranded themselves) as therapists, which although a common misconception is about as accurate as saying that computer scientists are software engineers (i.e. sometimes, but its not a one to one relationship). Thanks for the discussion, I enjoyed it.
I see where you're coming from, it's maybe useful to psychology practitioners. (I'll come across as a troll if I started airing my view that psychologists and psychiatrists are fraudsters so I'll stop there!).
"OK, I know a bit about pedagogy, and a lot of it is crap."
Any hard empirical evidence for this assertion?
"There's an awful lot of relativism in most of it."
Well, yes, people are not Turing machines or well defined physical systems. As Peter Medawar reminded us, science is the 'art of the solvable'. We need an 'art' for the things that are not solvable in the way he meant.
"...you know you're not in a field which cares about hard facts..."
Yes, teaching and learning is not a field that depends on hard facts alone. The 'soft' factors are important as well. Student teachers need some pointers, models, methods, examples of how to deal with the soft factors. Situated learning big time.
"...but they learn on the job and are pretty good after about 2 years..."
In the UK, most teacher training is based around a supervised teaching practice already, and our government is planning to make it more so. You do not become qualified until after your probationary year is complete (that is on top of the minimum one year training). Another strand is reflective practice, actually looking at your own teaching, its impact (or lack of impact) on students and how to change things. That means a student teacher can 'learn on the job' more effectively.
There's no hard empirical evidence that Scientology is crap, but I'm comfortable making that statement too.
Statistical studies showing little to no positive correlation between level of training (not including practicums - which the UK is obviously using extensively, good on them) and student outcomes is pretty damning though. There's some correlation between doing a masters level education course, and English teaching ability, but OLS shows this is just because students with good English ability (high writing scores in high school) are more likely to do a masters level teaching course.
> Yes, teaching and learning is not a field that depends on hard facts alone. The 'soft' factors are important as well. Student teachers need some pointers, models, methods, examples of how to deal with the soft factors.
For example, they should be learning psychology that's not so outdated that the Cosmopolitan magazine should be embarrassed to write articles on it (Myers-Briggs is totally discredited). I'm not saying I'm a hard reductionist. Education should be based on fundamental psychological foundations if possible, and while that's not always possible they should at least try to either teach stuff which is fundamentally sound, or actually works in the classroom.
"There's no hard empirical evidence that Scientology is crap, but I'm comfortable making that statement too."
I tend to agree. We both accept that there are grounds for belief that do not depend on empirical evidence or a scientifically articulated theory subject to strict Popperian falsifiability. I would extend that acceptance to include the idea that works of psychology, psychoanalysis, sociology, management theory and aspects of cultural studies and history may be useful to training teachers when trying to make sense of what happens in lessons!
"For example, they should be learning psychology that's not so outdated that the Cosmopolitan magazine should be embarrassed to write articles on it (Myers-Briggs is totally discredited)."
They don't in the UK. Gardner is quite popular though as a guide to lesson planning. Totally is a strong word though. Have a look at...
It does sound like teacher training in the US requires some looking at if the courses are not based around a supervised teaching practice. In the UK, during the one year PGCE course, the training teacher must teach for 150 hours under the supervision of a placement mentor (i.e. an actual teaching member of staff in the school or College). The University tutor conducts developmental teaching observations, and the placement mentor conducts the summative assessment (i.e. is person ok to let loose in my classroom). Our dear government is going even further and cutting the University element. There are doubts as to the wisdom of that.
Myers-Briggs as a psychological test is basically useless. MBTI indicates bimodal distributions of the attributes (people are either introverts or extroverts), when they are more like a normal distribution.
Gardner also lacks any real credibility, unless you throw out everything except "different people can be good at different things". It's good to encourage students to study a range of stuff (art, music and interpersonal stuff, not just math and english), but that's not really what Gardner is about. Students who are good at music aren't likely to learn math better if you sing it (OK, maybe I'm exaggerating).
Individualization isn't a great method. It's hard to do, and has a fairly small impact on student outcomes. Most students work best if you show them a diagram, explain everything, give them a chance to try it themselves, and demonstrate things in a number of ways. It's not about "different strokes for different folks"; most students like a bit of variety.
The reason many students struggle is because they lack the right foundations. They didn't learn it last year (for whatever reason) so they struggle this year and can't catch up. This is much more common than students simply not having the right talents.
I'm not from the US, but Australia. From what I can tell, the US varies from state to state. The problems with the US aren't really related to pedagogy, but social issues (large black / hispanic communities going to black / hispanic schools, where everyone is trapped in the poverty cycle) and the way their standardized testing dominates everything. Every year the teacher is mostly concerned about the students passing the end of year test, at all costs, and the end of year tests seem to be badly designed.
"Myers-Briggs as a psychological test is basically useless."
But as a story around the camp fire their theory (along with Honey and Mumford's development of it) might be useful. Stories, ideas, ways of thinking through. Basis for action, then evaluate the action. Kekule (benzene) and his opium.
"Gardner also lacks any real credibility"
But as a way of getting training teachers to think about the sensory modes they stimulate, there might be some value.
Action based research, situated theory, stories. Not science. Definitely not Science, depending on what time of day it is (before teaching, I did research)
Thanks. I'm a physics major, and I've done quite a lot of research into physics education, but I haven't looked into math education. I'll have to take a look into your references. I wrote this article a few months back when there was a flood of "why do we need math?!" articles after some editorial said it's worthless; I only got around to putting it up Friday. So it's aimed more to why we need math, rather than how we should teach it.
I've got a whole separate article on physics education which I'll have to put up soon.
"with techniques like problem-based learning, challenge-based learning, service learning, project-based learning, learning by design, learning through games, simulations, modeling, programming, etc."
I'm glad you mention games, because currently this field is mostly all about teaching arithmetic and other elementary subjects. Notable innovative exceptions are DragonBox and a few others in development.
It's probably a clichè to say that mathematics education is archaic, but it's true in a lot of cases. I've been to lectures where a simple 3-D program like Blender a few minutes of simple 3D work would have significantly helped countless students to imagine the mathematical entities being discussed. But alas, such things seem to be too advanced for much of academia - instead they build plastic models that only 1 person can use at a time.
I will present a somewhat controversial point of view here. The problem with math education lies as much with teachers as it does with students.
If you have students that cannot understand basic formulas and memorize them, you are not going to teach them abstract concepts. Concrete math is much easier than abstract math for the vast majority of people. I cannot really believe the stories about people who are supposedly talented at math but are not engaged enough by the school curriculum and consequently fail at the tests. School level math is very easy for anyone with some aptitude for math. Anyone who fails it will probably have even greater difficulties with abstract concepts behind it.
That gifted students are not stimulated enough is the real problem. Math classes for gifted children is probably the only solution. Something like Stuyvesant High in New York City but on a wider scale.
As for better math education in Singapore and Taiwan, well, they do have better students than most US public schools. Both genetics and culture play a role here. Asian people have both higher average IQs and higher average conscientiousness. Their teachers are able to teach math at a higher level. US could surely learn from them, but not every school could use their techniques.
I disagree. The current perspective of IQ and ethnicities show that while there might be slight differences, they are dwarfed by the differences in the education process.
If you want another example besides Singapore, look at french math textbook for highschool students - with nth derivatives, etc.
Here's the first link googles gives me - something that a finishing highschool student, around 17 years old, is expected to do if he is enrolled in a "scientific" curriculum. (basically there are 4 curriculums - scientific, economic, literary (humanities), technical (vocational))
I do not want to discuss race/national IQ differences, this is a sensitive topic and does not really belong here. I am quite sure individual IQ differences are not controversial though and cannot be "dwarfed" by good educational processes.
The French textbook looks reasonably rigorous but as you say it is intended for "scientific" students. Thats basically what I was talking about in my post - you need to select for gifted students before moving on to complex math (not that derivatives are particularly hard if you do not have to prove theorems). In US I believe differential calculus belongs to AP Calculus high school classes. No idea about their level but content on the internet looks similar, maybe less rigorous.
FWI, I do not believe in IQ. The evolution of scores alone makes me think there is a confounding variable. Students should just study what they can get money from and they are interested in.
In France, there are 4 curriculum of approximately similar sizes - at least when I did my studies. There is no IQ selection.
There is no selection either based on being "gifted" or not - you just get a curriculum matching the job you said you were the most interested in, if you basic grades are enough (i.e. if you persistently had below average grades in math, you might not be enrolled in a scientific curriculum)
"scientific" students makes 25% of the students - I'm not sure it can be compared to the situation in the US unless 25% of the students go in AP classes.
[The other curricumulum also have decent math, only slightly less theoretical or more into specific domains, such as arithmetic or geometric progression mathematics (for economy and some BTS vocational studies)]
As stated, this is a meaningless claim. Do you mean you don't believe in the accuracy or social value of of IQ testing, or don't believe that IQ exists as a measurable quantity, or don't believe in the practice of ranking people based on intelligence?
First, I don't believe that IQ exist as a measurable quantity - IQ is too many things thrown together in a single bad. Tests group together various things which may be handled by different subsystems in the brain. If there is something called IQ, it's an aggregate.
Anyway, if there was such a unique quantity, actual data shows a progression of IQ scores in time - therefore if the tests are considered accurate and unbiased, it must be a "quantity" that can evolve based on the society a person lives it. It should then be considered not as a value, but as a function depending on a variable called society.
I do not know if studies have tried to measure the accuracy of repetitive measurement in low education adults enrolled in a learning program. If there are such studies and if they show inconsistent result (ie any change of IQ), then IQ should be considered as a function of 2 variables : f(society, personal experience).
Now, even if we consider that at a time t it could be accurately measured and that societal bias could be removed, considering how other qualities (such as determination, work ethic, consistency, creativity, competitiveness...) influence the outcome of any human activity, it seems foolish to rank people based on just one quality - especially if we don't know the other values, and their individual ponderation in the end result.
This ponderation could also be different depending on the activity, and IQ provide an absolute advantage in some activities (rhetoric?), but say determination would give an absolute advantage in other activities (startups?).
I prefer to say I "don't believe in IQ" because it's easier to say that way than giving this long version.
> I prefer to say I "don't believe in IQ" because it's easier to say that way than giving this long version.
Yes, but for this subject, saying it that way is completely uninformative, to the degree that it's misleading. It would be like saying Harry is not now beating his wife -- it leaves too many questions unanswered.
Once again ColinWright graces the front page of HN on a weekend by submitting a story on mathematics education. The blog post submitted here, by an undergraduate physics major at the University of Texas at Austin, prompted me to read some of the author's other writings. The author's perspective on the importance of mathematics as a tool for understanding physics immediately reminded me of some good reads by older authors on physics. "How to Become a Good Theoretical Physicist" (HTML title "Theoretical Physics as a Challenge") by Nobel laureate Gerard 't Hooft
lists essential knowledge that everyone should possess who desires to advance theoretical physics, and included in that knowledge is much mathematics. There is a whole book, The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose,
that is marketed as a book about physics but includes a huge section reviewing secondary school mathematics as an essential background to physics.
The blog post submitted here has a title that is an homage to the article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by Eugene Wigner in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
People who know physics have long been delighted to find in physics applications for the mathematics they learned in mathematics courses without a hint of how useful the mathematics would be. The blog post author, however, goes beyond that perspective to urge, "Let’s think of mathematics in the abstract. Mathematics, at its most basic, is a very simple set of very well-defined rules. The rules describe the behavior and interaction of certain completely imaginary objects. Upon these rules, mathematicians have built others." And that brings to mind Paul Halmos's article (with its intentionally provocative title, an example of Halmos's spicy style in expository articles about mathematics) "Applied Mathematics Is Bad Mathematics" Halmos, P. "Applied mathematics is bad mathematics." Mathematics tomorrow (1984).
Halmos claims that mathematics is interesting and beautiful whether or not it has an apparent application.
Other replies already posted to this submission have helpfully mentioned the issue of empirical tests of what method of teaching mathematics may best help young learners appreciate (and later apply) mathematics. I have been deeply interested in cross-national comparisons of educational practice since living overseas beginning in 1982. In those days, one way in which school systems in most countries outdid the United States school system, economic level of countries being comparable, was that an American could go to many different places and expect university graduates (and perhaps high school graduates as well) to have a working knowledge of English for communication about business or research. I still surprise Chinese visitors to the United States, in 2012, if I join in on their Chinese-language conversations. No one expects Americans to learn any language other than English. Elsewhere in the world, the public school system is tasked with imparting at least one foreign language (most often English) and indeed a second language of school instruction (as in Taiwan or in Singapore) that in my generation was not spoken in most pupils' homes, as well as all the usual primary and secondary school subjects. At a minimum, that's one way in which schools in most parts of the world take on a tougher task than the educational goals of United States schools.
It was on my second stay overseas (1998-2001), that I became especially aware of differences in primary mathematics education. I began using the excellent Primary Mathematics series from Singapore
for homeschooling my own children, and I browsed Chinese-language bookstores in Taiwan for popular books about mathematics as my oldest son expressed an avid interest in mathematics. I discovered that the textbooks used in Singapore, Taiwan (and some neighboring countries) are far better designed than mathematics textbooks in the United States. (During that same stay in Taiwan, I had access to the samples United States textbooks in the storeroom of a school for expatriates, but they were never of any use to my family. I pored over those and was appalled at how poorly designed those textbooks were.) I discovered that the mathematics gap between the United States and the top countries of the world was, if anything, deeper and wider than the second-language gap.
Now I put instructional methodologies to the test by teaching supplemental mathematics courses to elementary-age pupils willing to take on a prealgebra-level course at that age. My pupils' families come from multiple countries in Asia, Europe, Africa, and the Caribbean Islands. (Oh, families from all over the United States also enroll in my classes. See my user profile for more specifics.) Simply by benefit of a better-designed set of instructional materials (formerly English translations of Russian textbooks, with reference to the Singapore textbooks, and now the Prealgebra textbook from the Art of Problem Solving),
the pupils in my classes can make big jumps in mathematics level (as verified by various standardized tests they take in their schools of regular enrollment, and by their participation in the AMC mathematics tests) and gains in confidence and delight in solving unfamiliar problems. More schools in the United States could do this, if only they would. The experience of Singapore shows that a rethinking of the entire national education system is desirable for best results,
but an immediate implementation of the best English-language textbooks, rarely used in United States schools, would be one helpful way to start improving mathematics instruction in the United States.
The blog post author begins his post with "In American schools, mathematics is taught as a dark art. Learn these sacred methods and you will become master of the ancient symbols. You must memorize the techniques to our satisfaction or your performance on the state standardized exams will be so poor that they will be forced to lower the passing grades." This implicitly mentions another difference between United States schools and schools in countries with better performance: American teachers show a method and then expect students to repeat applying the method to very similar exercises, while teachers in high-performing countries show an open-ended problem first, and have the students grapple with how to solve it and what method would be useful in related but not identical problems. From The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999): "Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x
"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77
A great video on the differences in teaching approaches can be found at "What if Khan Academy was made in Japan?"
I assume the answer is because all comments must begin with words, and those words seemed like a nice beginning. Maybe you should say what seemed notable about that sentence?
I disagree on the premise that math is tought in the wrong way. Repetition is a very important part of learning so learning to use mathematical constructs is just as important as learning to know how to build them. Also, people don't naturally think in abstract terms, many learn from example first, interpolate and then gain an understanding. Humans are more inductive than deductive. So at the very least it's not that the current education is wrong, but that it needs to add the teaching of another skill.
Like another poster says, the topic requires research. Teaching to think in abstractions isn't easy.
I don't recall ever wondering "why" I would ever "need" mathematics. I just always enjoyed it. To me, trying to convince students why they should enjoy mathematics (just because we do) seems like a good way to make students miserable.
Is mathematics education really failing because it doesn't produce many mathematicians?
I think this is the real issue. A small fraction of people are fascinated by mathematics and enjoy it. The idea that there is some specific way to teach mathematics that will ignite some spark in a person who doesn't inherently have it I believe is false.
This is indeed a serious problem. An interesting startup that is trying to address it is http://www.mathalicious.com/. They give math teachers lesson plans which tie math to things that students care about - 'is Kobe Bryant a better shooter than LeBron James?'. Things like that.
From my own experience, I was pretty bored with Math in high school. Through a series of coincidences I started reading some popular physics books, which led me to study Physics and re-discover Math as an incredible tool.
A pattern i see a lot is that people develop and interest in maths outside what their school was doing. Maybe we need to shift the onus onto individual children and families to foster interest in maths, regardless of how the schools teach it. It doesn't really matter how well you prepare the food if people don't have an appetite.
> You say that parents should take a role in their children's intellectual development? Then what am I paying property taxes for?
Let me ask another question -- do you really want to avoid having to provide an intellectual context for your children's life experience, and assign that responsibility to governments instead? Doesn't that sound dangerous? I shall resist quoting historical examples in which governments became the primary source of ideas and intellectual content for a new generation.
This article really should reference Kirby's excellent article about a non-Platonic concept of mathematics, which, in arguing that the writing of mathematics is necessary for the doing of mathematics. On this view, mathematics education is a necessarily in scriptural activity.
Kirby, Vicky. "Enumerating Language: "The Unreasonable Effectiveness of Mathematics"" Configurations, 11, 3 (2003).
A few years ago I read an article [1] about a school teaching maths by pretending that the process was magic in Harry Potter. Perhaps what children and adults lack is a sense of imagination of how to apply concepts.
Great article. The lego air plane analogy is very accurate. It is certainly true that maths has lost some of its mysticism in early education but I think perhaps telling students that "maths is a set of abstract made up rules governing an imaginary universe" might alienate them just as much as before.
great article, very well written with many agreeable points. But I doubt that the teaching of mathematics can be easily subverted into a better one, mostly because there is plenty of people that do not care enough: they think "math is hard and sucks" or simply do not care about knowing the alphabet of the knowledge. That's why at the University math is taught the good way, because there almost everybody is genuinely interested in the subject.
You seem to be unaware of how curriculum has been designed in recent decades. The current "Where's the Math" protest movement is complaining that the curriculum is devoid of calculation.
There's a few problems with the article. The rules argument he makes could instead be made for the more useful field of computer programming.
Also, the reductionist paradigm is dead. It worked for finding laws of physics and some chemistry, but otherwise, in the most useful fields of today, like biology/materials/chemistry (high throughput methods), computer vision, search engines etc. the automated creation of hypotheses through machine learning is winning out.
He uses the term mathematics, but completely removing that term and replacing it with computer programming and machine learning should be done. They sound cooler and this step will remove all the baggage and ill-will mathematics has in society. That way, all mathematics teachers can be simply removed from high schools and colleges and replaced with computer programmers. Estonia is one of the few enlightened places on this subject with their introduction of programming as mandatory from year 1 in k12.
Almost the entire high school drop out rate can be attributed to maths teaching, and their is no solid empirical proof of it's utility over programming (the practice of "science" hasn't been applied to "maths" itself).
http://en.wikipedia.org/wiki/Mathematical_anxiety
The same argument can be made for a number of other subjects, such as chemistry (why teach the periodic table?), languages (we have google translate and should taxpayers be funding this unproductive activity), and others (everything should be reexamined in light of search engines and instant information access).
The only useful subjects in k12 are probably
- computer programming, including machine learning/statistics
- physical education,
- communication (reading/writing/public speaking/socialization)
- general studies (trivia like history, geography, astronomy),
From reading your HN comments, you seem like the kind of person who has a small axiomatic set of principles, and then attempts to fit the world to those principles. Without going into the merits/demerits of that approach, I'd at least like to point out that it occasionally leads to absurd conclusions - such as replacing language courses with Google translate. Do you really think that Google translate is a feasible way to converse with someone in a foreign language? Or, for translating literature? Here is the opening paragraph of War and Peace, computer translation vs human. Note that Google translate assumed the whole thing was French, and left the Russian parts untranslated, so I had to manually translate the Russian words. Italics represent words originally in Russian.
Google translate:
Еh, my prince. Genoa and Lucca are only appendages of, the manor, Buonaparte family. No, I warn you that if you do not tell me that we have war, even if you allow yourself to overcome all infamies, all the atrocities of this Antichrist (my word, I think) - I do you know more, you are no longer my friend, you are no longer my faithful servant, as you say. Ну, hello, hello. I see that I frighten you, sit down and talk.
Richard Pevear and Larissa Volokhonsky translation:
Well, my prince, Genoa and Lucca are now no more than possessions, estates, of the Buonaparte family. No, I warn you, if you do not tell me we are at war, if you still allow yourself to palliate all the infamies, all the atrocities of that Antichrist (upon my word, I believe it) - I no longer know you, you are no longer my friend, you are no longer my faithful slave, as you say. Well, good evening, good evening. I see that I'm frightening you, sit down and tell me about it
Is it the responsibility of the taxpayer to fund the education for the leisure activity of the upper middle classes (speaking with foreigners or reading translated literature).
The education system that exists today (k12 + college) is a McReplica of what aristocratic people of earlier centuries considered fit for their children (who would never actually need to work). We're living the enviousness of the middle classes of a century ago (this same principle applies to the McMansions). It has no relation to what a more perfected reality could be.
With respect to google translate, I think we'll see a lot of improvements. I expect to see human level performance within 5 years (it's a "let the computers run long enough" + "enough data" problem).
>Almost the entire high school drop out rate can be attributed to maths teaching, and their is no solid empirical proof of it's utility over programming //
Can you point out the parts in the Wikipedia page that wouldn't apply if you teach programming and machine learning.
I'm not quite sure if mathematics is a superset of computer programming or will prove to be a subset. Either way we probably want to teach people algorithms to handle basic numerical functions (addition/addition/addition and addition) - why do you think it will matter if we call this learning by a different name? We'll probably want to take more of an abstract computer scientific approach than a "programming" approach that would need to focus on the stack to some extent, once you make those abstractions then you've got pure arithmetic haven't you?
In chemistry the periodic table is the means to categorise chemical knowledge isn't it. A sort of learning technique?
Languages I feel are useful but that state support should probably concentrate on major languages to foster cooperation between people groups, cultural understanding, mobility of workers.
I like your K12 subject list and appreciate the focus on utility.
You don't need the teaching of abstract concepts. Why not just let students roam free with "programming", a set of api's, so they can roam with free with creativity. Why even teach arithmetic? Just given them the arithmetic computer functions and let them play around with it. As students build games or whatever else they'll discover the concepts that you might consider important themselves.
What use is the periodic table for almost anyone. Teaching the periodic table should be left for someone wishing to pursue research in chemistry, maybe as a 20 year old. The basic skills that chemists probably use today, like computer programming, would have been ingrained throughout k12.
I find the idea of replacing "mathematics" in the curriculum with "computer programming" rather baffling. As important as programming is in our world, only a small fraction of high school graduates are going to need to do what we currently think of as "programming", while everyone will need to interact with technology and understand the world quantitatively. That calls for a much broader view of math and technology education than just "programming", even including machine learning and statistics.
Also, expecting anyone to have any kind of understanding of machine learning and statistics without a real grasp of what is currently considered "mathematics" doesn't make any sense to me. You're going to need algebra and linear algebra at minimum, and probably calculus as well, to have a chance at meaningful use of ML models.
Why do people need to know the under the hood stuff. As I mentioned in my other comment, we have the concept of division of labour.
Also, programming and machine learning will increasingly be most of what people's job description. In fact, in my extremest viewpoint, I think even programming is on it's way out and machine learning will the majority of people's work. Or put more simply, giving computers a few examples.
Already, with things like the Baxter robot from Rethink Robotics, and mechanical turk from amazon, we're seeing what the future of work will be like. The computers will just figure out things extrapolating from the choices we make.
I actually agree with you that programming may be on its way out for the majority of people. I can't agree with sweeping away all of mathematics, including arithmetic and algebra, as "under the hood stuff" for K-12 students.
Algebra (the art of rearranging equations) is not necessary anymore (you just need to know how to realize how things can be quantified - the naming of variables of computer programming). Arithmetic beyond use of calculator is useless.
Maths as traditionally taught is the leading cause of high school dropouts rates and probably a lot more damage than tahat (people who might have been scientists turning to arts instead).
> Algebra (the art of rearranging equations) is not necessary anymore ...
False, and false.
* As to the first, in common use, algebra is the abstraction of mathematical values and operations. If we can abstract a value to a symbol, we then can operate on the symbol rather than the value, greatly increasing the generality of mathematical ideas. Which operation is more useful: sqrt(2) or sqrt(x)? Another example -- which is more useful: D(7^n) = 7^n * log(7) or D(x^n) = n * x^(n-1)? (The first can only be applied to 7, the second can be applied to any values of x and/or n.)
* As to the second, algebra is much more useful now than it was 100 years ago. Those who don't learn algebra are inviting politicians to lie to them, which they will certainly do if given a chance.
> Maths as traditionally taught is the leading cause of high school dropouts rates ...
This confuses cause and effect. If we drop math education to prevent dropouts, we will have implicit dropouts (people who drop out but without physically leaving) instead of explicit ones.
> ... people who might have been scientists turning to arts instead ...
One's preference for art over science has deeper roots than a lack of comprehension of mathematics. And again, it confuses cause and effect.
I'll just have to disagree with you there. Your definition of algebra (beyond the naming of symbols, or variables in programming parlance) is not useful to most people. Where are they going to apply log or sqrt in everday life or even equations as you outlined (in my viewpoint, machine learning will automagically figure out equations or they will be apps on your smartphone).
I think people would be much better served implicitly dropping out than explicitly. People's identity has more to do with performance than innate ability.
Many people are mistakenly led to believe science is about symbolic equation rearrangement on paper (high school maths) + rote memorization and so choose arts instead. For many people, but not all, this is the reason why they don't pursue science careers. If they understood it was much more about teamwork and physical setting up and running equipment/software and thinking creatively, I'm certain we'd see different choices made.
But you aren't disagreeing, you're trying to redefine algebra based on personal preferences, in other words, you're arguing for the ascendancy of postmodernism (the idea that there are no shared truths, that everything is a matter of opinion and outlook). The problem with arguing from a postmodern perspective is that the argument instantly self-destructs (it applies first to itself, which means if you're right, then I have the right to ignore you).
The essential foundation of dialogue is consensus, shared ideas, and shared word definitions. Without those, there's no possibility for dialogue.
> Your definition of algebra (beyond the naming of symbols, or variables in programming parlance) is not useful to most people.
Utility is not the basis on which words are defined. Words mean what they mean, how we feel about those meanings is irrelevant.
> I think people would be much better served implicitly dropping out than explicitly.
So those who are qualified to be in school can be surrounded by those who aren't?
> If they understood it [science] was much more about teamwork and physical setting up and running equipment/software and thinking creatively ...
That's not what science is about, unless plumbing is about wrenches. You're confusing the practice of science with the idea of science.
I'm not doing some post-modernism or whatever sneaky jab you're throwing in.
You're implying that you have some definition of algebra that is truer than mine. But that is all meaningless arguments over words anyway.
The real issue is that I'm arguing that the vast, vast majority of people only need to know that concepts can be "quantified" - "variabilized" or "symbolized". This is mostly so that they can interact with computers for the purposes of computer programming, or for creating tables of data to be fed into machine learning black boxes.
You've also think you know a truer definition of science than me. I think science today is about collecting data and having computers make predictions, increasingly use black box machine learning. I'm inferring that you think it's about theorizing from data and writing out simple equations. I have disagree with you on that (though you'll probably turn around with your post-modernist jabs).
> I'm not doing some post-modernism or whatever sneaky jab you're throwing in.
Corespondent insists that there are definitions on which different people can agree.
> You're implying that you have some definition of algebra that is truer than mine. But that is all meaningless arguments over words anyway.
Correspondent insists that there aren't definitions on which different people can agree.
When you sort out which position you're taking, post again. :)
> You've also think you know a truer definition of science than me.
There is only one definition of science. Postmodernists, of course, disagree. For a postmodernist, there are as many definitions as there are people to have them.
> I think science today is about collecting data and having computers make predictions, increasingly use black box machine learning.
That is not how science is defined. You've just described how people collect data for the purposes of science, but science is defined by the uses to which data are put, not the data itself.
> I'm inferring that you think it's about theorizing from data and writing out simple equations. I have disagree with you on that ...
Feel free. But you need to realize that there are ideas on which people agree,and science is one of them. Indeed, without agreement about what science means, there can be no science -- science requires consensus about its own meaning, even while inviting disagreement about specific scientific theories.
reply to lutusp below because reply link takes too long:
You're just saying my definitions of things are wrong etc. etc. You just want to disagree with anything I say. My views are laid out clearly on the comments in this thread. I don't see what your views are (except to disagree with anything I say).
edit: replying to lutusp
You keep arguing that I'm just personal opinionating and that I'm wrong. I can't figure out what your views are - maybe I'm really dumb and you're really smart. You win.
> You're just saying my definitions of things are wrong ...
Locate where I said that.
> You just want to disagree with anything I say.
No, I want you to grasp some sense of reality -- not virtual reality, the other reality.
> My views are laid out clearly on the comments in this thread.
Yes, but your views are not compelling on what science is, as just one example. You expressed the belief that science's definition depends on personal opinion, which is false.
> I don't see what your views are ...
Then read. I explained what's wrong with defining science any way we please. All you need to do is read the words.
I'm curious how you would replace mathematics with programming. My undergraduate thesis project involves a lot of programming, but I also had to determine that I can detect anomalous gamma spectra through some linear algebra, a correlation matrix, an assumption of Poisson-distributed counts, and a chi^2-distributed anomaly statistic. This gives me a statistically well-founded way to detect anomalies.
Would you have me replace this process with some sort of automated machine learning, hoping it could produce better results than a special-purpose algorithm?
Even if your argument is true, that maths is needed for that particular task, it doesn't justify teaching it to 99% of people. Also, the math you mention could have been taught as a set of computer programming functions (taught in a few hours) rather than how it's traditionally taught (as s semester long class or sequence of classes).
Black box machine learning might have given you better method than your whatever you're currently doing. And it would be done in a automated manner, with human effort just focusing on collecting data. I'm assuming whatever you're done is being done/has been done by many 1000s people as well. That's a tremendous waste of time. Even if the method you outline is better than what machine learnig would have found, it only needed to be done once (or a few times - marketplace) and exposed as an api.
The point of my thesis is that it has not been done by anyone else before. In fact, I had to do the mathematics because previous work in the area required too much training data to be effective for mapping; it is simply not viable to require dozens of training passes through an area before I can begin to detect anomalies.
My algorithm works because it uses assumptions from the physics of the problem -- the assumption of Poisson-distributed counts.
Well, I can't argue with the specifics of your problem since I don't know about it (but the general point that this doesn't justify the current maths system for 99% of people).
However, at some point in time, computers will be taking over all human abilities (if you believe that AI is inevitbale). At what point will that occur? It's my strong suspicion that we have already reached that point and any evidence to the contrary is simply that people aren't trying automated methods (I believe that machine learning can solve any problem that humans can solve, and furthermore, this wouldn't require more computing power than we already have).
That's missing the point entirely. I'm pretty sure you don't know the exact materials and the process for making the shoes you might be wearing. There are probably no shoe factories left in america. They same applies to the black box machine learning, very very few people need to know of what goes on under the hood.
edit: I don't know if you changed your comment, but black boxes exist that can solve any problem (neural nets for example).
That might be true for some very general (or common) problems (though I have doubts about that as well), but it can't possibly be true for a specialized problem such as the one described. How do you train a model if you don't understand the problem? You can't just say "unsupervised learning" and expect your algorithm to come up with a full understanding of the physics involved.
I agree with you, I just feel like that is basically the understanding of variables (names of concepts) and not maths. So, you need to understand what measurements to take/can be taken, and then set up that system and feed it to the black box machine learner.
Now, there still need to be some people who understand how things work in detail, but this is much smaller than what is being taught (most k12 do not need to know maths or physics or chemistry etc). They do need practice in abstract thinkng and this would be much better done with programming (a playful, creative, logical, and useful activity).
The reason why math is 'hard' is the same reason 'exercise' is hard. It takes a combination of skill and practice (a/k/a discipline) to make it 'easy'. And in a similar way to the endless "how to lose weight" fads, we see "how to do make math easy," when <easy> is not 'in their nature'.
The main problem with math education is that it is falsifiable. Math teachers can be measured, because math performance can be measured. This is why it makes educators nervous. And, I suspect, why they mostly suck at teaching it. They are the ultimate risk-averse sub-population. Subconsciously, 'educators' <hate> the idea of accountability, because it is predicated on the notion of potential failure. But the 'educational' establishement is <built> on the foundation of unquestionable <authority>. so math is for these guys an existential threat.
TLDR: To be good at teaching math, you have to be open to measuring your failure, which makes people feel bad. Including the teachers.