kitchen table math, the sequel: Physics Education and Failures in Conceptual Understanding

Thursday, July 2, 2009

Physics Education and Failures in Conceptual Understanding

For decades, physics professors in universities and colleges in the US have known there is something wrong with physics studies in their schools. Their own physics majors have a poor understanding of basic concepts in mechanics, electricity and magnetism, and quantum mechanics.

Journals like The American Journal of Physics (devoted to teaching and pedagogy at the university level) and Physics Teacher (the same for high school and lower) bring up these issues, with a variety of proposed changes and solutions both at the individual classroom level and at the higher theory-of-ed level. D. Hestenes at ASU and his colleagues have done work in this area, both in questioning the failures of pedagogy and developing some solutions. First, the problems.

D. Hestenes wrote in "What Do Graduate Oral Exams Tell Us?" (Am J. Phys. 63:1069 (1995)) of finding a quote from physicist W. F. G. Swann, in "The Teaching of Physics", (Am. J. Phys. 19, 182-187 (1950)):
"Much can be said about oral examinations for doctor’s degrees, and in my judgment not much can be said that is good. I have sat in innumerable examinations for Ph.D. at very many different universities, sometimes as a member of the permanent faculty and sometimes as a visitor. In almost every case the knowledge exhibited was such that if it represented the true state of mind of the student, he never should have passed. However, after the examination is concluded there is usually a discussion to the effect that: "Well, So-and-so got tied up pretty badly, but I happen to know that he is a very good man," etc., etc., and so finally he is passed."


Hestenes goes on to quote Swann as saying [A student] "passes his tests frequently [including graduate comprehensive exams], alas, with very little comprehension of what he has been doing."

Hestenes diagnoses the problem as this:

It seems not to have occurred to the faculty that dismal oral exams may be symptoms of a severe deficiency in the entire physics curriculum. I submit that there is good reason to believe that they are symptomatic of a general failure to develop student skills in qualitative modeling and analysis.


These general failures mean that even students who have the grades to appear to have excellent mastery of the material do not understand basic elements of the material they have "learned".

It also suggests that college students who fail to understand the material may end up there because their confusion prevents them from attaining the mastery the "good" students have.

Of course, the errors didn't just start in college. Generally speaking, proper physical intuition is lacking in students who took high school physics, even in those who did well. Hestenes writes in "Force Concept Inventory", (Physics Teacher, Vol. 30, March 1992, 141-158)
"it has been established that1 (1) commonsense beliefs about motion and force are incompatible with Newtonian concepts in most respects, (2) conventional physics instruction produces little change in these beliefs, and (3) this result is independent of the instructor and the mode of instruction. The implications could not be more serious. Since the students have evidently not learned the most basic Newtonian concepts, they must have failed to comprehend most of the material in the course. "


Hestenes et. al. wrote the Force Concept Inventory, a multiple choice test whose aim is to "to probe student beliefs on this matter and how these beliefs compare with the many dimensions of the Newtonian concept. " It poses questions that force a choice between the correct Newtonian answer for an explanation of a given system, and other commonsense explanations that are actually misconceptions. After the test, interviews are done to determine students' reasoning.

Here's an example of a misconception that the FCI aims to tease out of a student:
[The misconception of "impetus":]
The term "impetus" dates back to pre-Galilean times before the concept was discredited scientifically. Of course, students never use the word "impetus"; they might use any of a number of terms, but "force" is perhaps the most common. Impetus is conceived to be an inanimate "motive power" or "intrinsic force" that keeps things moving. This, of course, contradicts Newton’s First Law, which is why Impetus in Table II is assigned the same number as the First Law in Table I. Evidence that a student believes in some kind of impetus is therefore evidence that the First Law is not understood.


The FCI has been given to thousands of college and high school students. The above paper details the results on the FCI, given as a pre and post test to both high school and undergraduate physics courses, with tremendous detail on similarities and differences across classrooms in the country. More, it provides strong evidence that traditional college physics pedagogy isn't doing anything to teach physics to the students who take it:

"The pretest/post test Inventory scores of 52/63 for [The Regular Physics Mechanics course at Arizona State University] are nearly identical to the 51/64 scores obtained with the Diagnostic for the same course...we have post test averages of 60 and 63 for two other professors teaching the same course. Thus, we have the incredible result of nearly identical post test scores for seven different professors (with more than a thousand students). It is hard to imagine stronger statistical evidence for the original conclusion that Diagnostic posttest scores for conventional instruction are independent of the instructor. One might infer from this that the modest 11% gain for Arizona State Reg. in Table III is achieved by the students on their own. "


Which brings us back to the state of physics majors going to graduate school:

One of us (Hestenes) interviewed 16 first-year graduate students beginning graduate mechanics at Arizona State University. The interviews were in depth on the questions they had missed on the Inventory (more than half an hour for most students). Half the students were American and half were foreign nationals (mostly Chinese). Only two of the students (both Chinese) exhibited a perfect understanding of all physical concepts on the Inventory, though one of them missed several questions because of a severe English deficiency. These two also turned out to be far and away the best students in the mechanics class, with near perfect scores on every test and problem assignment. Every one of the other students exhibited a deficient understanding of buoyancy, as mentioned earlier. The most severe misconceptions were found in three Americans who clearly did not understand Newton’s Third Law (detected by missing question 13) and exhibited reading deficiencies to boot. Two of these still retained the Impetus concept, while the other had misconceptions about friction. Not surprisingly, the student with the most severe misconceptions failed graduate mechanics miserably, while the other two managed to squeak through the first year of graduate school on probation.


Is it just that the Chinese students who manage to get into US physics grad schools are such creme de la creme that they are perfect? Or is Chinese instruction vastly superior?

(And for those who wonder about American instruction in other subjects, read this and weep:)

One disturbing observation from the interviews was that five of the eight Americans, as well as five of the others, exhibited moderate to severe difficulty understanding English text. In most cases the difficulty could be traced to overlooking the critical role of "little words" such as prepositions in determining meaning. As a consequence, we discarded two interesting problems from our original version of the Inventory because they were misread more often than not.


And yet, those who make it through physics graduate school to professordom mostly correct these errors, at least in mechanics. (Though not necessarily. In quantum mechanics, new professors are notorious for teaching elements of the material incorrectly. In special relativity, David Mermin, prof at Cornell, believes many professors teach the entire subject wrong. (He discusses this in a paper called something like "how to teach Special Relativity.") Hestenes suggests this is due to the realities of post quals grad school: the day in, day out teaching and researching refine one's intuition over and over again.

I think this implies something else as well. Error correction in intuition can only occur and stick if the mastery of the manipulation of the equations is so strong that you (correctly) believe what they tell you. If you can be forced to do the math on the board, and forced to read and think about what it says, then you can learn the truth counter to what your intuition tells you, but only if you are utterly sure you did the math on the board correctly.

If instead, you doubt yourself, doubt your manipulation of equations, doubt your application of the laws as you understand them, then you will get confused, doubt your answer, default to your intuition, and scrap learning the correct way to think.

That means you need a tremendous amount of mastery. How in the world to achieve that?

Hestenes' answer --changing how physics is taught in high school and in college--will be explored in a week or so.


Physics Education and Failures in Conceptual Understanding
Fixing Physics Education: Modeling Instruction
Physics Education Continued
More Modeling Instruction: Techniques

27 comments:

OrangeMath said...

Earlier your readers were critical of Physics First. The arguments for it are very strong - against pretty weak by relying on "tradition." The following pages may help:
http://www.aapt.org/aboutaapt/updates/upload/physicsfirst.pdf
http://orangemath.com/648/ http://orangemath.com/3477/

SteveH said...

The following is from:

"http://www.aapt.org/aboutaapt/updates/upload/physicsfirst.pdf"


"The upper-level class (second-year physics course)
can be more rigorous because students have a
good foundation on which to build."

Physics First is not about rearranging the order of science courses, it's about expecting kids to take more courses. I should hope that taking more courses is better than taking fewer courses.

Wouldn't be nice if K-8 schools prepared kids properly in math and science so that this extra course isn't necessary.

Allison said...

This is a group blog. Some of the folks here asked about Physics First, and others were critical. I knew nothing, and so I've been investigating. I hope to present a fair appraisal of what Hestenes' modeling theory does right and wrong so that we can perhaps get to some more refined distinctions in what works in inquiry-based teaching and what doesn't.

I do agree that there is a tendency on this blog to have skepticism about anything constructivist, and I
think it's well founded. But any discussion of constructivism has to acknowledge SteveH's consistent point that Low Expectations drown out the supposed constructivist or traditional methods anyway. Constructivism can't work in the low expectation model in most classrooms in this country, no matter how well constructivism works in special multi-hour, multi sensory rooms. Hestenes' classrooms are far from those low expectations.

But to give a preview to where I'm going: while I will agree that Hestenes' modeling instruction has definitely improved physics understanding over traditional methods for many students, there remain a lot of questions about why. It's not like there's been a regression confounding variable to pull out the teacher knowledge from the inquiry based part right now--though maybe data is moving in that direction. The more I read about modeling instruction, the more that teaching skill makes massive differences: quality of teaching matters enormously, the thoughtfulness and care with which the teachers explain the answer (yes, some of them give the correct answers first so that students are primed to go in the right direction), and the ability to empathize with students and understand their misconceptions. It may be that the constructivism of physics first isn't the reason for the success of the FCI, but the enormous attention to fixing teacher create misconceptions is.

SteveH said...

There is also the issue of prerequisites and sequencing. Science in K-8 is a vast unknown wasteland. High schools typically have an unknown starting point in terms of science and math. Physics First might be better given the circumstances, but it might be better to fix the circumstances.

There were lots of things I didn't like about my "traditional" education, but any sort of rigorous discovery education in high school would have to pass a detailed sniff test. I'm not sure I could trust K-12 schools to do it properly.

I'm looking forward to Allison's review of Hestenes' work. My initial reaction to one of the web sites about modeling theory was not positive. That was mostly becuase I couldn't quickly get past the happy talk to the details.

Tracy W said...

I wonder how this fits in with what Dan Willingham writes about the difference between expert knowledge and a novice's knowledge - that as far as we know novices have to go through a stage of imperfect understanding where their knowledge of the subject is inflexible - that no one has discovered a way to go straight to deep understanding.

Paul B said...

I would put forth the proposition that problems with physics have a root; problems with math.

We are very fortunate in my district to have lots of very talented career changers teaching math. My (mandated for career changers) masters class has about 15 teachers in it; all from my district. Half the class is career changers with extensive practical depth in mathematics in their former lives and education(engineering, meteorology, etc.). The other half of the class is comprised of teachers who are great at math but lack practical application of it.

Here's the point (finally). For all but the last two courses you could have walked into one of our classes and it would not be obvious who was who. For our last two classes, both calculus, you would be able to classify these groups quite readily.

If you've experienced entry level calculus you know that its problem sets are rich in dynamic physics. The equations of motion are steeped in the calculus. It provides a superior tool set to algebra when dealing with motion.

Here's what happened in my calculus classes. The career changers waltzed right through it while the others hit a wall. The wall they hit was not in the grinding out of a derivative or integral, not the calculus. The wall was in knowing how to set the problem up.

Knowing how to sketch the 'nut', knowing how to assign and document variables, and knowing how to discern the most appropriate path to a solution were the stumbling blocks. The people who struggled looked like they were struggling with the physics inherent to calculus problems. They weren't. Calculus is the first time you put all the fundamentals into the game and physics is the field you play it on. The strugglers were playing their 'first game', the career changers were playing their hundredth.

None of these problem sets were posed as physics lessons and they were always amply demonstrated by an excellent teacher, yet these folks, who were very good at math, struggled mightily. Why? They'd never had enough practice doing this stuff. Calculus is kind of like the grand unifying theory of mathematics. It requires expertise in everything that precedes it and one of the best ways to give it life is physics problems.

The career shifting engineer has hundreds of these problems under their belt. This kind of exposure creates habits of thinking and documenting that facilitate solutions. Someone who just learned about derivatives a month ago (even though they may be very good at doing them) is not prepared to set them up from all the nefarious ways that rich problem sets can disguise this root simplicity.

We (career changers) could usually help the strugglers by exposing just a few initial steps. A light bulb would go on and they would be off to the races grinding out a correct solution.

I submit that this ability comes from simple practice; lots and lots of practice. Modern education provides little of it. Every time I walked out of these classes this year I would think, "Wow, I'm doing this to my kids. I'm racing through a proscribed curriculum like the white rabbit and they never get enough practice to make it part of their core."

This is the real danger of curricular bloat. Every new fad displaces practice. It's not rocket science. It's why I get kids who write illegibly. It's why I get kids who can't multiply, divide, or even add.

They know what a dodecagon is and they can tell you what a mode is but they can't add! I would argue that kids who struggle with physics probably have little idea what a fraction is or how to manipulate it either. I would argue that they probably don't know how to work with scientific notation, or logarithms, or how to multiply by a power of 10.

Physics instruction on top of weak math skills, so overloads the ZPD as to make its pedagogy meaningless. You can't play basketball if you can't dribble.

SteveH said...

"...as far as we know novices have to go through a stage of imperfect understanding ..."

AP Physics requires: Newtonian Mechanics, Electricity and Magnetism, Fluid Mechanics and Thermal Physics, Waves and Optics, and Atomic and Nuclear Physics."

Students are unlikely to be ready to understand electricity and magnetism from the standpoint of Maxwell's equations, so they have to focus on some level of conceptual understanding and some level of ability to do solve problems.

I expect that there are many ways to improve or reduce the stages of "imperfect understanding", but I don't see that coming from Physics First. It may be a better sequence for many kids (defninitely not for those who are prepared), but it does it by adding another course, not fixing the overall K-12 science and math curricula.

Given a choice, K-12 schools will center all courses around hands-on group learning in class. I'm not talking about one day a week labs with very specific goals. I'm taking about minimal transfer of knowledge from the teacher to the students. This doesn't define discovery. I have had many light bulb discovery moments doing individual homework and during direct instruction. I distinctly remember how I felt when I was directly taught about integration as the area under the curve. I also remember exactly how I felt when I learned how path integrals worked.

To me, discovery in K-12 is really about group learning in class, which I consider to be extremely wasteful of time. As I've said before, only a few kids (if any) get the light bulb experience and then proceed to explain it (poorly) the the other members of the group. Some kids still won't understand, but it will all look like a happy learning environment.

So, launch those marbles and plot the data, but I really need to see the big picture of how this is one step in a path that leads to a first principles understanding of Newton's laws of motion. Hopefully, it will be one step that doesn't ignore the poor and inconsistent K-8 teaching of science and math.

I also hope the discussion can focus on more than a trivial discussion of traditional versus new. I didn't like my traditional high school science classes. I had calculus in high school and started out as a physics major in college. I think that one of the reasons I didn't last long was that there was no clear path to help me transition through the different levels of inflexible understanding.

I think it has to do with what Allison was talking about...

"...utterly sure you did the math on the board correctly."

Even though I was good in math, I still did not have the skills or knowledge to derive everything based on first pricnciples. This has much more to do with math than it has to do with conceptual understanding. I was never taught how to do that. At that level, I think that understanding derives from the math, not the other way around. There was no clear path back to first principles. This often requires very good math skills (at least through differential equations), and students generally don't get that until their sophomore year in college.


After I transferred to engineering, I think I eventually worked my way to the ability to understand things from first principles, such as applying beam theory to many structural problems. I can't say I know how this happened, except to explain it by brute force. However, I don't think it has to be this way. It will be interesting to see what Hestenes has to say about it, but how much can you do in high school?

However, this discussion is far removed from Physics First and what goes on in a typical K-12 classroom. I hate to talk about these things because K-12 educators will spin it in all sorts of ways.

So, what conceptual understandings or modelings in high school can help one better make the transition to a mathematical approach to first principles in college?

SteveH said...

"...problems with physics have a root; problems with math."

I think I would agree that this applies at all levels of understanding.


"I submit that this ability comes from simple practice; lots and lots of practice."

I would agree with this too. There is a tendency for educators to look for the magic pedagogical bullet. However, I remember many times when I thought I understood the teacher's lecture, but then proceeded to really learn the material during the 40 problem homework set I had to do.


"Every new fad displaces practice. It's not rocket science."

That's the problem with this discussion. I would like to talk about real discovery and about learning topics from first principles, but the real question is why kids get to fifth grade not knowing what 6*7 is. I don't want to change the subject.

bky said...

The marble-tossing reminds me of one of the best and most memorable labs from my freshman college mechanics class. We dropped marbles in front of a grid and photographed their fall with a strobe light. From this we calculated the gravitational constant. Utterly cool.

To do this we needed (1) a little physics, (2) calculus for the equation of motion, (3) calculus to make sense of the least-squares fit that we did (zero gradient of a certain sum <--> least squares fit), as well (4) as enough algebra to actually be able to perform the calculus required for the physics and the statistics.

Paul B said...

Kids can't do 6 times 7 because they are 'taught' multiple ways to multiply. Then they discover the one they like best (usually involves fingers) and stick to it until somewhere along the way they are given a calculator to expedite things.

They are assessed subjectively. They are promoted without regard to those assessments. And finally, they're never given targeted remediation or practice that is up to the task of keeping them on track.

Other than that there's not much wrong.

ChemProf said...

Allison said: "It may be that the constructivism of physics first isn't the reason for the success of the FCI, but the enormous attention to fixing teacher create misconceptions is."

This is the experiment I would like to see. My class is a classic lecture, plus I am the campus Luddite, so barely use any of the much ballyhooed technology, but I do spend a lot of time DIRECTLY dealing with student misconceptions and having them work problems so I can identify those misconceptions. In science education, it simply isn't enough to present the correct concept. You have to recognize common student misconceptions and stamp them out. Otherwise, perfectly bright students graduate with really appalling ideas!

"A private universe", the 80's video of Harvard graduates who can't explain the seasons is a good example, but I have a lot of them. One of the things that was interesting in "A private universe" was that once the correct concept was presented, students came up with really convoluted ways to incorporate the new, correct information with their earlier, incorrect ideas, coming up with something even worse!

I am coming to think this is one of the problems with the "multiple methods" of elementary math. It can introduce student misconceptions, and if those aren't cleaned up quickly by using rapid feedback, they become ingrained and nearly impossible to remediate.

Paul B said...

I had a student once who had this bizarre method of division. He would cover the page with tick marks, hundreds of them. I watched and discovered that he was 'ticking' off the divisor amounts while 'ticking' up the answer. It worked really well for fanciful problems like 32 divided by 8.

Unfortunately for him I needed him to problems like 327.43 divided by 38. He was very unhappy with me.

While his method was a fundamentally sound whole number treatment it sucked at anything big or decimal. You could see that he had adopted a comfortable (likely) demonstration method. It took me a year to get him off of that train.

The sad part was that he was excellent at multiplication, estimating, and subtraction; all he needed to master division. This bright kid was actually deathly afraid of division as his misconception had taken over.

ChemProf said...

Absolutely! A lot of these methods start to fail with "weird" numbers. One thing I had to learn not to do is give examples with "easy" whole numbers. Often, students can easily do a problem with 2 moles of something but can choke on the same problem with 1.277 moles where they can't just see the math.

All of this is bringing me back to Engleman's direct instruction. Wasn't the point of that to avoid accidentally introducing misconceptions, especially when you couldn't guarantee that every teacher had a sufficiently deep understanding of the material to avoid such introductions and fix them when they arose. After all, my fourth grade teacher and I had a big argument about where the seasons came from -- she thought it was the planet moving further from the sun!

Allison said...

Paul and Steve,

The problems with physics graduate students' lack of correct intuition is NOT a math problem at its root. They are simply too good at the math, and too good at the problems they correctly work.

There's overwhelming evidence that the problem is a physics problem at its root, and that practice--as defined by doing every problem in, e.g. Halliday and Resnick/Sears, Zemansky,Young/etc -- DOES NOT fix it. I will find get some of the AJP papers on this subject for us, and you will see. You will see that you yourself may have some very bad physics conceptions, or at least you may have corrected them very late in life without ever noticing how far you've come.

The reason why practice isn't enough is because we're practicing THE WRONG THINGS. We're practicing the problems inside the wrong model in the first place, so our practice reinforces our errors. ChemProf is 110% right about this; it's like the Bohr-atom model. If you think there's an electron zipping around a nucleus, then you can still get As in chemistry all the way up to your prelims by creating convoluted ideas of what that electron's doing.

I'll get some concrete examples of the misconceptions up in another post before I go on. Really, don't pre judge this stuff. Wait and see, I think you'll be interested.

And it does get back to Physics First. Really, because the Physics First people are actually trying to get a set of conceptual understanding in high school that ARE correct so the transition to a mathematical approach in college makes sense.

Doug Sundseth said...

I suspect these fundamental problems in the understanding of physics may arise from the way physics is usually taught. Specifically, in physics instruction, ontogeny recapitulates phylogeny; the history of nearly every significant error in understanding is taught, in historical order.

There is some value to this method; if nothing else, it teaches the scientific method moderately well. But it may be that teaching all these errors causes students to remember some of the errors too well, and not as errors.

Is there really a need to teach the plum-pudding model of the atom or phlogiston theory any more? For that matter, teaching Newtonian physics (other than as a useful approximation of low-relative-velocity relativistic physics) is teaching error.

F != mA

(Absent significant clarification that is not usually provided until quite late in the sequence of instruction.)

Anonymous said...

This is in response to OrangeMath, who said, "Earlier your readers were critical of Physics First. The arguments for it are very strong - against pretty weak by relying on "tradition." The following pages may help:
http://www.aapt.org/aboutaapt/updates/upload/physicsfirst.pdf
http://orangemath.com/648/
http://orangemath.com/3477/"

I visited the first website, and OrangeMath is correct to call this "arguments." Not research. The closest I see to research is this:
"In 2001, Project ARISE issued a report of a research project conducted by Spencer Pasero on the state of Physics First programs.3 Fifty-eight public and private schools were studied. The schools that responded were all teaching physics as the first course and most have reported that this inverted sequence has been successful. Since this study, several large urban public school districts have joined the larger number of private schools by offering physics first and the institutions have met with mixed reviews."
http://www.aapt.org/aboutaapt/updates/upload/physicsfirst.pdf

"[R]eported that this inverted sequence has been successful" is not evidence of anything; It is just an opinion, and a not-very-informative opinion at that. What does "has been successful" mean? That it is superior to traditional methods? Almost as good? Detrimental to test scores, but kids and teachers feel better using it?

Not impressed, OrangeMath

OrangeMath said...

The last Anonymous had a good point: lack of evidence, but somewhat weak, because he/she gives no evidence that the current system is effective either.

Research evidence is from ACT on the correlation for Physics, Chemistry, and Biology for graduation college, with no correlation for Earth Science/Integrated Science. However, most/many students don't take the three cores. They stop after Biology. This can be interpreted as 1) weak students taking non-core science in high school are being zapped by US, which may be immoral on our parts; and 2) delay Biology until other courses are already taken (a bit sneaky, yes) to maximize students graduating college.

The logical argument for Physics First/Capstone Biology is that Physics can be done with simple Algebra F=MA, Chemistry with A + B -> C + D, and Biology with a strong understanding of Probability and 3-D Models from Chemistry.

The lack of awareness and prevalence of mistakes in Physics is a separate problem. The development of mechanical feel takes time. The offshoot of PSSC, www.sci-ips.com, is one way to develop this in Middle School.

Finding meaning is the bugaboo. Cognitive Science only points to the difficulty of this. Read Dan Willingham's book for this. It's somewhat mandatory for discussions with teachers now IMHO.

OrangeMath said...

Of course test scores in Physics First is the killer and the REAL reason Physics First cannot be adopted: a bunch of ninth graders will test poorly against a select group of juniors and seniors. It's about politics, not learning. Research misleads on this point to the point of intentional deception. There is no substitute for using your head!

Physics Last started with the wonderful Physics efforts of the first half of the 20th century. Once, the helix was determined, molecular biology became the most important science - that's really not a difficult to justify opinion.

High School Biology should be an involved course, not the simple, stick-figure course it is. Physics needs to take it rightful place: first in time, not last in line.

ChemProf said...

There you go, SteveH: Physics First is better because more students take physics so it is more democratic. And because it improves biology instruction (if they actually build on the chemistry, which is a big "if"!) Nothing to do with students having a better understanding of physics by any measureable criteria.

By the way, that I'm not knocking Conceptual Physics or disagreeing with Allison's basic point that there is a problem with traditional science education. But I don't see that taking a more conceptual approach to physics means that you should teach it first, necessarily. Nor is a good grasp of Newtonian mechanics going to help you much in Chemistry class.

rocky said...

Just to be contrary, I'd like to stick up for starting simple and slowly adding complexity. When we first teach about motion we use things with constant velocity: trains. Later on we introduce projectiles with constant acceleration and no friction. Then we might add a crosswind, or wind resistance. Later we might have rockets which lose mass as they go, so they get faster, faster.

My point is, kids and people learn by starting off with a simple universe. Euclid started with a ruler and a compass. The Singapore bar models work because the numbers are easy. Johnny gives away three fifths, and then gives away half of what he has left. The good teacher uses these pictures to show his kids how to turn words into mathematical expressions. When they can label the small box "x" they are ready for algebra and can handle problems with large common denominators that no one wants to draw.

Of course kids should know six times seven. It may not be the answer to life, the universe and everything, but it should be memorized. They should know long division, and short division with single digit divisors.

They also know prime factorization and how to get a GCF or LCM, how to reduce fractions to simplest terms, how to add, subtract, multiply and divide them, and how to convert between improper fractions, mixed numbers, decimals and percents.

They should know how to turn word problems into math equations and how to set up simple direct proportion rate equations:

feet = (feet/second) * feet
gallons = (gallons/mile) * miles
grams Oxygen = grams water * (grams Oxygen/grams water)

I'm sure all this is uncontroversial. I just wanted to say it because if they can do these things, I am not worried if they believe z-orbitals look like little hourglass shapes.

rocky said...

Oops (red face).

feet = (feet/second) * seconds

A teacher friend of mine says don't use girly fractions (with the slash). Use Manly fractions (with a horizontal line), because it is easer to read what's on top and what's on the bottom.

SteveH said...

I think the discussion is getting a little bit confusing. All of this started out with an article I saw about Physics First. My complaints did not deal with whether reordering the sequence of science classes was a good thing to do or not. The presumption is, however, that what physics loses, biology and chemistry gain. I don't see this happening.

I see Physics First as being quite different. Schools not only rearrange the order, but they shift the classes down a year. The schools still offer their old advanced science courses and claim that the new, reordered, and shifted down classes are conceptual feeder classes to the more advanced classes.

So now, ALL freshmen have to take a conceptual physics class, many of whom haven't had algebra yet. They talk about hands-on group class work and we ALL know what that's about. I can't imagine that K-12 educators magically get concepts, discovery, and experimental science right in 9th grade.

If they want to claim that adding in an extra conceptual course before the real course is better than what they had before, then I might not disagree. (I still might think it could be done much better.) However, this is not just rearranging the order of the classes. It's adding another course.

Then Allison brought in a different, top end view of the problem and refered to Hestenes' work. Although Hestenes talks about high school, I don't see anything about Physics First in his work. Both methods might talk about a better approach to concepts, but I don't see any other connection. Are some schools applying Hestenes' work in a Physics First environment? If so, then Physics First ends up being too vague a topic without seeing the details. I like his idea in science of "Follow the Energy", but a lot of what I've read is NSF grant proposal blather. I'll use a different adapted quote: "Show me the details".

Even though my traditional science education in high school was far from perfect, it didn't make me stupid. I'm also convinced that there is no pedagogical magic bullet. I see any sort of new approach to conceptual modeling as helping the best students who are already good in math, not the others. It reminds me of Project Lead the Way. It doesn't matter how many concepts the kids learn if they stink in math. It doesn't matter how much the kids like the material if they stink im math. Motivation and concepts will not fix bad math. Pie chart concepts of fractions do not directly translate into a mathematical understanding of manipulating rational terms. Non-mathematical concepts might be a good place to start, but I don't see them as magic solution to the hard mathematical work yet to come. After the very lowest level, I would expect to see concepts, math, and rigor at each level. Also, I would like to see the concepts-only level go away by 8th grade at the latest.

Michael Weiss said...

The high school in which I used to teach opened for business in 2000. As a young, private school we had a clean slate and a free hand to implement curricula as we saw fit. Our science department head went with a Physics First curriculum for the following reasons:

Physics First is a "logical sequence", in that Year 1 ends with the atom (including the wrong-but-useful Bohr model), preparing the ground for Chemistry in Y2, which begins with valence and bonds. Then Y2 ends with organic molecules, preparing the ground for Biology in Y3, which starts with photosynthesis, respiration, and other processes involving those organic molecules. In Y4 students then have the option of revisiting any of those three courses in an AP version.

It made good sense, but also had some serious problems: specifically, a large chunk of our 9th graders were taking Physics without first having taken Algebra. This made equations of motion almost unteachable.

In my last year there the math and science departments spent some time working together on a coordinated math/science curriculum. There would have been two tracks:

1. Students who passed Algebra in middle school would take "Algebra-based Physics" at the same time they took Geometry. The two courses would be synchronized so that, for example, the Geometry teacher would cover vectors and then the next day the Physics teacher would use them; the Geometry teacher would teach basic trigonometry and then the Physics teacher would use it (for example to break vectors into horizontal and vertical components).

2. Students who had not yet had Algebra would take "Conceptual Physics" at the same time they took Algebra. There would be a similarly coordinated approach between those two courses.

In either track, the two courses would serve each other's interests -- the science course would provide an opportunity for math practice, and the math course would make sure that the science teachers could rely on their students having the necessary mathematical prerequisites when they arose.

The plan would have continued on into 10th grade, although the details were never worked out. We knew there needed to be some coordination between, for example, teaching logarithms in Algebra 2 and studying pH in Chem.

Unfortunately none of this was ever implemented. The reason? It was too complicated in a school our size to offer two different versions of every science class and to coordinate students' schedules so that math and science were aligned.

Michael Weiss said...

Note by the way that "Physics First" does not necessarily mean constructivism, group learning, or any other pedagogical strategies. I think it's a mistake to lump them together.

SteveH said...

"...Physics First does not necessarily mean constructivism, group learning..."

Don't tell us. You need to tell that to many of the schools implementing it. I don't think they would know how to do it any other way.

Michael Weiss said...

I really don't understand that.

Like I said, my school did "Physics first". It had to do with the sequence of courses, and with the sequence of topics within the courses. Not with teaching methods.

DeeDee said...

Internationally, is it more typical to teach high school sciences sequentially, or integrated throughout the highschool years?

Here in Ontario, both maths and sciences are taught as integrated courses through both grades 9 and 10.

Math continues to be integrated right through grade 11 (geometry, algebra, trig and statistics) which is the final year of math required for matriculation. In grade 12, students can choose a variety of courses, including a fifth math credit which includes calculus.

Science, as I said, is integrated in grades 9 and 10, and those courses include topics in each of biology, chemistry, physics and earth/space sciences. In grades 11 and 12, all four streams are taught as separate courses (earth sciences is typically only available in grade 11, all others are available in both grade 11 and 12) , so that science-oriented students can take as many as 7 additional high school credits in their final two years. A typical course load is 7 or 8 credits per year, so it's possible for math and science together to account for more than half of the grade 11 and 12 credits for students who are headed into sciences in university. Only one grade 11-12 course is required for matriculation, however. (Some schools also offer additional specialized courses in environmental sciences or engineering as well as these standard courses.)

AP courses are not that common here, but where they are taught they are typically an honours-stream course replacing the standard grade 12 course in that subject, often requiring an accelerated grade 11 course as a pre-requisite.

DeeDee