more precisely how do we change the curriculum? and how do we ensure that all schools have enough access to computers to use them as a teaching and learning tool
The singular way to do it would be to employ math teachers with math/science degrees, not (just) teaching degrees. I had 3 teachers in high school with a math/science PhDs (I went to a magnet school), and their teaching was heads and shoulders above all the other teachers...save one, who I believe had a Master's in physics.
This means I disagree with madmaze. Change the teachers, and the curriculum changes will follow. Change the curriculum with the same teachers, and you get New Math < http://en.wikipedia.org/wiki/New_Math >, which had some good ideas but was doomed.
Problem is, math and science degrees are a lot more expensive to employ.
I agree with the need for actual degreed people in Math and Science teaching. If an adult went to a training center and got a person with an education degree and certificate instead of someone who put the years in for a technical degree, you would want your money back.
The bigger problem is that you need to pay Math and Science degreed people more than other teachers because of scarcity and other high paying job opportunities. The current educational machine won't allow that.
I believe that there are currently more significant problems than those mentioned in the (very good) TED talk---how do we stop the continued trend of denigration of teaching and the profession in general? The current populist idea of blame all of the teachers for the nightmare that administration gone insane is, is not going to lead us unto the future. It will lead us back to a time of ignorance and idiocy.
Populism favors common people and blaming the elite. The idea you describe ("blame all of the teachers for the nightmare that administration gone insane is") is the opposite of this.
Just as you don't really understand how a CPU works until you've programmed in assembly, you don't really understand math if you haven't done hand-computation, at least up to a point. If your approach to any math problem is "plug it into Mathmatica" you will never be able to creatively apply basic concepts to more complicated problems and you will never have any sense of whether the answers it provides are "reasonable."
That said, tools like Mathmatica are here to stay, and the argument that we need to spend more time teaching the use of these tools as an aid to exploring and understanding fundamental concepts is probably correct.
To expand on you're point: math (and some math-like subjects) are flat out different than driving a car.
I think the problem with your example, ams6110, is that Wolfram might retort that software developers don't need to have programed in assembly. Yes (says Wolfram), the absolute best developers in the world might be well-versed in the this, but that doesn't mean all developers should be so trained, in the way that we train everyone to do arithmatic.
To respond to this, I would say: no one would ever study Ramanujan theta functions without learning linear algebra, and no one should study linear algebra without learning arithmetic. Most other subjects must be compartmentalized, because it is impossibly difficult to derive biology from basic chemistry; some things must be accepted as given. But math is different. In math, you really can derive everything from the ground up, and doing so is immensely valuable.
> Just as you don't really understand how a CPU works until you've programmed in assembly, you don't really understand math if you haven't done hand-computation, at least up to a point.
Conrad Wolfram addresses this very point in the video (11:30). He says that the most valid point that hand-calculating helps understanding is that teaching processes themselves helps understanding, but that there's a better way to do that: programming. In other words, programming a math solution is better for teaching math understanding that hand solving it.
In the end, I think the idea he's trying to impart is that the core of math is not calculation at all, it just seems that way because they've been intertwined for so long. But now that we have computers to do the tedious calculations, we can work on defining and exploring what math really is: how different conceptual intuitions are related
Agreed, but only up to a point. To program it is fundamental to have a strong intuition as to what should happen, why, and how to get there. Doing that without some amount of manual computation is temerary.
You may be correct but no one starts out programming in assembly then working their way up to higher order languages.
However this is exactly how math is practiced.
Wouldn't it be better to start with higher order logic first, so students can see a value proposition then work our way down to the nitty gritty computation that no one needs to do unless they're advancing the field?
I started programming with circuits, and then machine code.
And I think you're wrong about starting with higher order logic. It's too easy to make plausible statements that are simply wrong. Learning the real nitty-gritty is essential to avoid sweeping errors.
A good portion of this problem would be solved in the long(er) term if we pursued a 3-track plan. First, cut back on the amount of bureaucracy running the school systems. Whenever state cuts roll around, you never hear about office administrator's getting cut. . .just teachers. Where I live, we have an absurdly high ratio of administrator to teacher ratio and when cuts come around the teachers go first. The more overhead we've got, the less we can spend on teachers to retain the people who actually teach.
Second, give the schools more autonomy to choose curriculum and meet the needs of their students. Instead of standardizing the curriculum across the state and federal level--shoehorning every student, no matter their educational needs into one specific mold--give the schools more latitude to meet their students needs. Give the teachers some room to choose what they teach instead of having them teach straight to the standard of learning tests.
Finally, end teacher tenure and promote/fire based on merit. Good teachers who teach well should get raises and encouraged to stay teaching. Teachers who stink (I know I can think of several off the top of my head my classmates and I universally agreed stunk) should be let go and replaced with new teachers. Overtime, this builds a a stronger cadre of motivated individuals with an incentive to do well teaching instead of having protection from the all powerful teacher's union to
As an engineer, I love statistical data or numbers I can crunch and use to make a supposedly more objective comparison between things. I believe the problem with standardized testing is we are trying to take that idea and then apply it to something inherently non numerical--the acquisition of knowledge. Contrast this with economics and I think there's a startling difference. In Econ, we participate with numbers, something easily quantifiable. In education we participate with our minds and we have no easy, objective, way to quantify this into convenient numbers. For instance, standardized testing is inherently biased against people who don't take tests well meaning the test will not accurately represent their level of knowledge which could then have further repercussions when they apply to college.
The problem is we are looking at this through an engineering lens. When looking for a quality teacher, other factors than results on a silly test should play a role. How do the students respond to the teacher? Is the teacher usually well prepared? Do they have a good reputation with the students for being fair? Are their lectures interesting? At some point, on the local level (individual school), I think parents and the administration have a good idea of who the good teachers are and who are lemons and I don't know if you could quantify this into a statistical system that is "impartial".
This has been submitted many times in the past few months, with the same point and counter-points rehashed endlessly. Do a search for Conrad Wolfram to see what I mean.
I'm not going to repeat my arguments. For some people he's right, for others he's very wrong.
How is math applied? Well with his four steps he has a start, but it's not very good.
For "computation", it is not clear if he means arithmetic, algebraic manipulation, or both.
People should learn to do some arithmetic by hand if only as a start on better understanding; then, of course, they should use calculators or computers when the need a lot of arithmetic or just don't want to worry about arithmetic errors.
For algebraic manipulation, in well done math, that's rarely a problem and is easily enough done by hand and is better done by hand because it aids understanding. Since the understanding is the main goal and the main difficulty, aiding understanding is good. Maybe he wants people to use some algebraic manipulation software, and that's fine but only for narrow and relatively rare applications.
He never gave a good description of what math really 'is'. Sad.
For "teaching" math, he is not clear if he is talking about K-12, college, graduate school, trade school, on the job training, or what.
For college and graduate school, his concerns about 'computation' are not accurate. E.g., if work through Rudin's 'Principals', then 'computation' will play no significant role.
In high school, in a well taught course in plane geometry, what math really is begins to be illustrated, and computation plays no significant role.
For our K-12 teaching system doing a good job teaching math, that's hopeless -- the subject is far too difficult for our our society to do that broadly.
So, if a student in K-12 wants to do well with math, then they need to get outside the K-12 teaching system and proceed with guidance from people who are actually good in math and from some good books and lectures.
The math is on the shelves of the research libraries and isn't going away. Only a tiny fraction of the population will ever get very good with math, and that has always been the case.
So, is there a real problem in teaching math in K-12 that can be solved within the K-12 teaching system? Not really.
Does the speaker really understand math and its teaching? Not really.
Ideas are fun when you're not the president. Most people would be far far more conservative if they had absolute power... or would quickly become so after a failure or two.
God says...
punishable detested absence deeps hap table While daughter
towards valiant This threatenest rude entrusted wonted
pertain certainty within gift If Orestes presently committed
proceed consuming sufficest reposed stablished feeds year
stirrest impurity air re Prophet functions undisturbed
dragging both consists annoyance masses meeting disappear
ceaseth purpose parents' residest SEND astrologer taxes
22 comments
[ 0.20 ms ] story [ 51.3 ms ] threadThis means I disagree with madmaze. Change the teachers, and the curriculum changes will follow. Change the curriculum with the same teachers, and you get New Math < http://en.wikipedia.org/wiki/New_Math >, which had some good ideas but was doomed.
Problem is, math and science degrees are a lot more expensive to employ.
The bigger problem is that you need to pay Math and Science degreed people more than other teachers because of scarcity and other high paying job opportunities. The current educational machine won't allow that.
That said, tools like Mathmatica are here to stay, and the argument that we need to spend more time teaching the use of these tools as an aid to exploring and understanding fundamental concepts is probably correct.
I think the problem with your example, ams6110, is that Wolfram might retort that software developers don't need to have programed in assembly. Yes (says Wolfram), the absolute best developers in the world might be well-versed in the this, but that doesn't mean all developers should be so trained, in the way that we train everyone to do arithmatic.
To respond to this, I would say: no one would ever study Ramanujan theta functions without learning linear algebra, and no one should study linear algebra without learning arithmetic. Most other subjects must be compartmentalized, because it is impossibly difficult to derive biology from basic chemistry; some things must be accepted as given. But math is different. In math, you really can derive everything from the ground up, and doing so is immensely valuable.
Conrad Wolfram addresses this very point in the video (11:30). He says that the most valid point that hand-calculating helps understanding is that teaching processes themselves helps understanding, but that there's a better way to do that: programming. In other words, programming a math solution is better for teaching math understanding that hand solving it.
In the end, I think the idea he's trying to impart is that the core of math is not calculation at all, it just seems that way because they've been intertwined for so long. But now that we have computers to do the tedious calculations, we can work on defining and exploring what math really is: how different conceptual intuitions are related
I completely agree. But couldn't this be said about education in general? Public schools seem to be stagnated in archaic methods of teaching.
Case and point, how many web developers do you know who could outline the TCP/IP protocols in great detail?
However this is exactly how math is practiced.
Wouldn't it be better to start with higher order logic first, so students can see a value proposition then work our way down to the nitty gritty computation that no one needs to do unless they're advancing the field?
And I think you're wrong about starting with higher order logic. It's too easy to make plausible statements that are simply wrong. Learning the real nitty-gritty is essential to avoid sweeping errors.
Second, give the schools more autonomy to choose curriculum and meet the needs of their students. Instead of standardizing the curriculum across the state and federal level--shoehorning every student, no matter their educational needs into one specific mold--give the schools more latitude to meet their students needs. Give the teachers some room to choose what they teach instead of having them teach straight to the standard of learning tests.
Finally, end teacher tenure and promote/fire based on merit. Good teachers who teach well should get raises and encouraged to stay teaching. Teachers who stink (I know I can think of several off the top of my head my classmates and I universally agreed stunk) should be let go and replaced with new teachers. Overtime, this builds a a stronger cadre of motivated individuals with an incentive to do well teaching instead of having protection from the all powerful teacher's union to
> end teacher tenure and promote/fire based on merit
These two points may not be compatible. How do you propose promoting based on merit other than the results of the standardized tests?
The problem is we are looking at this through an engineering lens. When looking for a quality teacher, other factors than results on a silly test should play a role. How do the students respond to the teacher? Is the teacher usually well prepared? Do they have a good reputation with the students for being fair? Are their lectures interesting? At some point, on the local level (individual school), I think parents and the administration have a good idea of who the good teachers are and who are lemons and I don't know if you could quantify this into a statistical system that is "impartial".
I'm not going to repeat my arguments. For some people he's right, for others he's very wrong.
Yes. The video clip gave some examples early on.
How is math applied? Well with his four steps he has a start, but it's not very good.
For "computation", it is not clear if he means arithmetic, algebraic manipulation, or both.
People should learn to do some arithmetic by hand if only as a start on better understanding; then, of course, they should use calculators or computers when the need a lot of arithmetic or just don't want to worry about arithmetic errors.
For algebraic manipulation, in well done math, that's rarely a problem and is easily enough done by hand and is better done by hand because it aids understanding. Since the understanding is the main goal and the main difficulty, aiding understanding is good. Maybe he wants people to use some algebraic manipulation software, and that's fine but only for narrow and relatively rare applications.
He never gave a good description of what math really 'is'. Sad.
For "teaching" math, he is not clear if he is talking about K-12, college, graduate school, trade school, on the job training, or what.
For college and graduate school, his concerns about 'computation' are not accurate. E.g., if work through Rudin's 'Principals', then 'computation' will play no significant role.
In high school, in a well taught course in plane geometry, what math really is begins to be illustrated, and computation plays no significant role.
For our K-12 teaching system doing a good job teaching math, that's hopeless -- the subject is far too difficult for our our society to do that broadly.
So, if a student in K-12 wants to do well with math, then they need to get outside the K-12 teaching system and proceed with guidance from people who are actually good in math and from some good books and lectures.
The math is on the shelves of the research libraries and isn't going away. Only a tiny fraction of the population will ever get very good with math, and that has always been the case.
So, is there a real problem in teaching math in K-12 that can be solved within the K-12 teaching system? Not really.
Does the speaker really understand math and its teaching? Not really.
God says... punishable detested absence deeps hap table While daughter towards valiant This threatenest rude entrusted wonted pertain certainty within gift If Orestes presently committed proceed consuming sufficest reposed stablished feeds year stirrest impurity air re Prophet functions undisturbed dragging both consists annoyance masses meeting disappear ceaseth purpose parents' residest SEND astrologer taxes