Cap'n Refsmmat

A four-dimensional spacetime manifold is a mathematical object. If you cut it out, you have lost the ability of your theory to make useful predictions. Because the observed characteristics of gravity are more complicated than that. See, for example: the precession of Mercury's orbit, black holes, gravitational lensing, etc. Conventional models of gravity were inadequate to explain these phenomena. Modeling space as a four-dimensional spacetime manifold predicts observations much more accurately. Call me back when I've taken a general relativity course. Because objects follow straight lines in space -- as you know from Newtonian physics -- and if space is curved, the "straight lines" aren't quite. Just like how straight lines on Earth, like longitude lines, are actually curved. At some point, however, asking "how" is pointless. I could answer by saying "All matter is made up of incredibly tiny gnomes which behave according to this set of mathematical rules. They conspire to prevent any experiment from ever directly observing them." You could argue that they are, in fact, elves, but we'd have no way of settling the dispute. One must be careful to restrict scientific discussion to what can actually be tested. I can observe that objects follow certain paths, and that these paths would be straight lines if we modeled space as a four-dimensional space-time manifold, but I cannot perform an experiment in which I sit the manifold down on a leather couch and ask it why exactly it feels obliged to behave this way. Intermediate particles are from quantum mechanics.

A friend and I wrote a series of programs in TI-BASIC to solve problems in our algebra class. We even programmed it to show its work, make useful plots, and everything. Sometimes I think it counts as cheating, but I justify it by pointing out that we had to understand the math to write a program to do it. The people we shared it with, on the other hand...

I see this all the time assisting for a physics course. Mid-semester, we passed out an anonymous survey in class to ask what students thought of the course, whether certain things were helpful, etc. Everyone complained that the homework did things which they didn't cover in class, as though the point of homework is to ensure you can memorize and repeat everything the professor said. Fortunately, our professor responded with this: Agreed. I think, though, that this is partly the point of interactive and inquiry-based classes such as the ones in the papers I linked to. They're halfway to the Moore method, I suppose. I think the current AP physics curriculum requires free-body diagrams. There's often a free-response question requiring the student to label the forces on a particular object in a problem. What I often see, though, is students confused about the normal force, or who draw the current velocity of the object as a force, as though it needs some impetus to keep it in motion. They refer to it as "the force of the object" or something like that.

There's a difference between a course actively promoting misconceptions and one which gives the students a poor conceptual understanding of the subject -- though they are quite capable of memorizing equations and grinding out answers -- and leaves them to fill in the conceptual gaps with their own intuitions and invention. I suspect we see far more of the latter. Students learn all about the physics terms, but have little understanding of how they connect to reality and how they can be applied to physical situations. If you'd like to see the examples from the research, here's the citation: McDermott, L. C. (1984). Research on conceptual understanding in mechanics. Physics Today, 37(7), 24. American Institute of Physics. http://physicstoday.org/resource/1/phtoad/v37/i7/p24_s1 If you can't get a full-text copy easily, I can send you a PDF. I don't think the blame can be placed solely on the students. Somehow, we're designing courses which students can pass easily with little conceptual understanding, and we're teaching them in a way that encourages them to simply memorize steps to solving problems. We could blame this on the students all we want, but how do we fix it?

I'm familiar with this method, as I'm typing this sitting in the R.L. Moore building here at UT. I wish I had an opportunity to experience it in my courses, but only a few math professors are willing to put in the time and effort to run a Moore-method class. The answers I see professors give to questions can be placed into several categories: An answer to a question completely different from the one asked. This is the most common. An answer given before the student can even finish explaining their question. Most professors start answering as soon as they hear something they can answer, without waiting a moment for the student to finish. (There is, incidentally, some research on waiting before answering. It prompts the student to elaborate and think about their idea some more, and elicits answers from other students, helping the entire class learn from one question.) An answer that requires knowledge of topics that have not been covered or defined. The professor doesn't seem aware of what the students do and do not know. A repetition of what was already said. The student is confused, and the professor just restates what's confusing. An actual answer to the question. The student might not understand it, but it's an answer. A ten-minute speech about things completely unrelated. (My ethics professor was terrible about this.) I don't see how self-esteem enters into it. Blunt answers aren't the problem. Useless answers are. And why do you think students are systematically failing to meet their responsibility? Is there something that can be changed to fix this? The studies were performed on students taking calculus-based university physics. I think a decent high-school teacher can produce students capable of easily passing the AP physics exam and passing exams in their university physics courses, but without the students understanding squat about basics of physics. A lot of the research shows that students come to classes with misconceptions produced from general experience, not earlier physics courses -- students think that a force is required to keep an object in constant motion, for example, despite being told about Newton's first law as the first subject in their physics class. Teachers may eloquently explain the correct concept, but misconceptions are robust: the student makes the explanation fit their beliefs, rather than throwing out their incorrect ideas. No amount of competence can lecture away the misconceptions; they need to be handled directly.

Here's a few examples of the papers I've found. They tell us interesting things about effective teaching methods: Improved Learning in a Large-Enrollment Physics Class found that interactive teaching methods significantly improved learning and student attendance in an introductory physics course. Learning and retention of quantum concepts with different teaching methods showed that in an introductory quantum physics course, interactive methods significantly improved conceptual understanding. Classroom demonstrations: Learning tools or entertainment? found that traditional lecture demonstrations are ineffective at teaching unless students are asked to make predictions beforehand. These studies show useful methods that undergraduate lecturers could use to significantly improve student understanding. We could blame educational failure on a student's inability to learn, but not until we've investigated our own teaching methods. Indeed. But the same strategies could presumably be adopted in secondary school to prevent the students from developing the misconceptions in the first place. Of course, that puts a tremendous burden on the teacher. Understanding the subject matter isn't enough -- you need the ability to understand the student's understanding of the subject matter, and that can be very difficult to do. It requires a thorough understanding of the subject and an understanding of how students learn it. So I suppose I do agree that we need better-educated and better-prepared teachers who understand their subject exceedingly well. But we also need to change our teaching strategies, because they're not nearly as effective as they should be.

A link to the original source would suffice.

Plagiarism is not permitted on SFN. Please cite your sources when you quote text verbatim.

They can't be effective in this responsibility if they are utterly incapable of explaining a concept to a student, or of answering questions helpfully. There's a great deal of research on teaching methods at the undergraduate level, particularly in physics, and the majority of it is ignored. Simple changes, like asking students to predict the outcome of a lecture demonstration beforehand or waiting more than three seconds before answering student questions, have been shown to have significant beneficial impacts on student understanding. There are also a number of studies following students in introductory mechanics classes and evaluating their conceptual understanding of the material. Students generally make horribly contorted and confused mental models of basic physics and fit everything they read or hear into the model. (One paper found that many students have difficulty distinguishing between velocity and acceleration -- after months of intro physics.) Once this happens, the student can't possibly succeed in learning for himself, until someone carefully demolishes his faulty model. I can provide a few papers if you'd like.

If those are indeed the derivatives, that makes sense. I presume you took the gradient and set its components to zero?

Here in Texas, high school science and math teachers are required to have an undergraduate degree in science or math along with their teaching certification. (Texas state schools don't offer education degrees.) There's no guarantee that you won't get, say, a biology major teaching an advanced-placement physics class (as happened to me -- he ended up being removed ten weeks into the year), but in general teachers have more competence than just the introductory science classes they were required to take for their education degree. If a state were to require physics teachers to have physics degrees and math teachers to have math degrees and so on, it would need essentially the entire physics major output of its universities just to provide teachers. That's not going to happen until teaching pays as well as the numerous alternatives. I'll also note that I've had quite a few incompetent lecturers in university, where they're nearly all tenured professors. I think we're missing something more fundamental than "knows math and science well." Do you think this is the cause or the symptom? If these students were taught algebra without calculators, but are now entirely dependent on them to do basic algebra, something went wrong in the pre-calculator teaching. If they were taught to do algebra by using their graphing calculators to type "solve()", then some teachers should be taken out and shot. Yes, our university is running a program to get science majors teaching certifications before they graduate, so they have an undergraduate science degree and are licensed to teach immediately. It's moderately popular and they're working to expand it.

We're not going to solve the problem for you. Can you show us what you've tried, or explain where you're confused?

Let me be clear: We are not in a position to help you. I suggest you seek professional help from someone qualified to help you, rather than the Internet.

I'm not trying to argue against using calculators, and I hope I didn't come across as attacking the students. They're surely capable of doing algebra and arithmetic should they have an opportunity to learn and practice. Somehow our education system isn't giving them that opportunity. The use of calculators to do basic algebra is merely a symptom of this. I agree. I'm concerned more about the system which doesn't successfully accommodate these differences. We're designing courses which students take and pass without gaining a fundamental understanding of the material. This is a problem with how we teach, not how they learn. How do we teach better?

Calculators are not permitted on a large portion of the AP physics exam. The AP exam is a standardized exam which many universities accept for college course credit: http://www.collegeboard.com/student/testing/ap/sub_physb.html

I just got back from tutoring students at a local high school. It was an event for students specifically taking AP classes, which culminate in an exam which is accepted for credit by many universities. The students are generally high school seniors -- 17 or 18. Every student had a TI-89 calculator which they used to solve every problem. At one point, a student was working out some arithmetic and encountered this: [math]2\times \frac{4}{5} =\; ?[/math] She paused, looked uncertain, and said "well, I don't know that off the top of my head" and reached for her calculator. I'd ask other students "okay, now solve that equation for t" and they'd stare at me blankly, before reaching for their calculator to type "solve(4.9t^2 = 10, t)". They had trouble with basic physics concepts. For example, a force that acts perpendicular to the direction an object travels does no work -- it doesn't contribute to the object's motion. (For example, if a block slides along the ground and you push straight down on the block, you're not doing work on it.) They were utterly surprised when their calculator gave them zero. Many didn't get the concept of inverse sines and cosines, despite having done trigonometry for months. You can see why I'm worried about the state of education. Most of this is basic algebra and arithmetic, and yet these high-school seniors couldn't do it without their TI-89s. They had passed their state standardized tests, passed their algebra classes, gotten into an advanced placement course -- and they can't multiply fractions. What can we, as people in the science community, do? What can educators do to change this? We can't just make the tests harder -- they'll all fail. There's something structurally wrong with how we teach high schoolers. I'm curious to hear if anyone else has had similar experiences as well.

How would you detect a frequency chirp in a neutrino detection? Neutrinos are very difficult to detect, so out of many pulses you may only detect one neutrino. With only one neutrino, you can't reconstruct any clever pulses.

iNow, you give other SFN posters too much credit. I don't believe any others have mastered your brand of sarcasm, mockery, and insult. Certainly no other members have achieved the impact you have in terms of reported posts, warnings, and staff member departures. However, I am intrigued by this comment: What are your goals here?

I'll refer you back to the second part of post 21 and swansont's addition. Furthermore, suppose you do formulate the "mechanics" by which spacetime guides matter and light into curved paths -- something beyond what general relativity provides. How will you test your hypothesis?

OPERA has now used this strategy to replicate their results: http://blogs.nature.com/news/2011/11/neutrino_experiment_affirms_fa.html They've used 3ns pulses to generate much shorter bursts, and have replicated the 60 nanosecond early arrival time for the neutrinos. Of course, this doesn't rule out problems with their timing equipment, so we'll have to wait for MINOS on that.

How could I experimentally determine this? The non-entity-ness of spacetime would not prevent me from making highly accurate predictions about the universe. Making predictions about the universe is all science claims to do. Perhaps I should make a clearer statement of this. Science is in the business of making models that predict the behavior of the universe. By "models," I mean mathematical and conceptual systems that can be used to describe reality. Science tests these models by comparing their predictions with actual experiments. However, it is not necessarily true that the components of the models -- such as various mathematical concepts used to compute their predictions -- correspond to physical entities in reality. No experiment could prove this. We can only demonstrate that reality behaves as though it were made of the components the models describe. For example, there's no experiment I can do to prove that spacetime is made of one kind of substance or another. I can, however, prove that the universe behaves as though it were made of a four-dimensional spacetime with certain mathematical properties. In short, then, science describes what the universe acts like. It does not describe what the universe "is", because that can't be determined empirically. (After all, I could say "the universe really 'is' x", but you could retort "no, it's a bunch of gnomes which behave exactly like x", and I'd be stuck. If two different explanations produce exactly the same results, how can I distinguish between them through experiment?)

My point is that the behavior of, say, spacetime is well-defined by the mathematics. A physicist could predict how it will behave in any situation by simply using the mathematics of general relativity. If I choose to specify what spacetime "is" -- what kind of entity -- I will not add any more information to our description of its behavior. Hence I won't be able to say "if it's ontologically this one kind of thing, it will behave differently, and I can test that in experiment." For example, I could specify that spacetime consists of incredibly tiny gnomes that behave following rules that exactly replicate the rules of general relativity. The gnomes cooperate to alter the image of any microscope capable of seeing them, so they are essentially undetectable. Since they behave according to the rules of general relativity, their behavior is completely indistinguishable from, say, the behavior of an organized clan of subatomic Roomba robotic vacuum cleaners which also follows the rules of general relativity. So unless the nature of the "entity" of spacetime actually changes its observable behavior, there's nothing for science to test. I didn't say I don't have to devise an experiment. I asked whether there is such an experiment. There isn't.

Depends. If I hypothesize what the distorted medium "is", can I test that hypothesis by experiment? This is a crucial point, so please don't dodge it.

Depends. If I hypothesize a specific relationship between mathematics and the material universe, is there an experiment I can do to test it? Yes. And it's a fairly poor metaphor. It essentially says "mass deforms space, and objects like to slide downhill because of gravity, and therefore mass attracts things." It uses gravity to explain how gravity works. Not very helpful. Science doesn't need "it" to be an actual entity; it needs "it" to be useful for predicting the results of experiments, whether or not "it" exists.

Whether or not it "actually" exists as a malleable entity won't affect the mathematical predictions of the theory, so it has no observable consequences. So in science, it doesn't matter.