Allison on Math Nights

MSMI has done several parent math nights.

When we are asked to give a math talk by a school and it is well attended, it is because the parents are upset. If it is very well attended, it is because the parents are in an uproar about the math program.

Since we generally are going in to fix the math program, or to support a math change to it, our goal is first to name the problem. We explain the issue (nationally, not just locally, not just here, wherever we are, but nationally) is that US curricula are not preparing kids for algebra. We tell parents what they know intuivitely but can't name. We tell them what they've watched their older kids suffered through. Then we explain we need to change what we teach, when we teach it, and what the teachers know about the maththey teach to fix it. When we are done, generally, parents calm down and give us the benefit of the doubt.

Usually, the second math night (a followup) has 1/4 of the turnout the first one had.

If a math night has no attendees, it is because math is doing just fine--the parents are concerned about some other problem.

Parents don't have time to go to meetings if things are fine. They go to indicate their disapproval or their concern.

We also found if the *children* put on the math night, as one of the grade night programs, it is well received--so if we want parents to learn about the math program, learn the games to practice math facts, etc. then it needs to be a child-centered event. Parents come when kids put on a math carnival. They even enjoy it. It does not need to be fuzzy math--kids LOVE stumping their parents at mental math calculations and bar modeling.

We tell parents what they know intuivitely but can't name.

Knowing intuitively that something is amiss: this is the chronic problem parents face. You know something--your cognitive unconscious knows something, rather--but you can't name it.

I remember, when I first became politically engaged here, living in a state of chronic anxiety that a) I didn't know what I was talking about and b) I was about to be publicly called out on not knowing what I was talking about. I spent hours Googling and reading, and reading and Googling, to make sure everything I said and wrote in my district had already been said and written by someone who did know what they were talking about.

Kitchen Table Math was incredibly important to that effort. I wrote posts to put into words what my cognitive unconscious already knew (or suspected), and I said nothing, in district, without ktm commenters vetting it first.

Funny thing: at some point I stopped feeling anxious, and I stopped obsessively fact-checking myself.

I hadn't become an expert on math or math instruction or public schools in general, but somehow I knew enough to feel confident that anything I said -- even something I said off the top of my head -- would be in the realm. Which it generally was.

I also, and I hesitate putting this in print, developed a sense of how thin my adversaries' knowledge was. That's not a criticism. Administrators can't possibly know everything about every subject (that's the problem with central administrators choosing math curricula), and an administrator who went to ed school before constructivism was in full bloom may not actually know that much about the doctrine and its history, however committed s/he may be to "rolling out" one constructivist initiative after another.

In short, at some point it dawned on me that I could pretty much say whatever I wanted and get away with it. I could get sloppy and no one would know but me.

That came as a bit of a shock.

I see politicians and pundits differently now.

Politicians and pundits are churning out an awful lot of content.

How often do they actually know that what they're saying is true?

How much fact checking happens in politics?

I'm guessing not too much.

Allison and Palisadesk on high-SES versus low-SES kids and schools

ALLISON:

Here in the Twin Cities, we are experiencing multiple and opposing forces at the same time.

Hainish, I see some low-SES kids in private schools here that are worse off than if they were in high performing low SES schools. The rest of the school is barreling along doing discovery math, and these kids have no chance to learn. High SES kids are eventually tutored privately, but low SES kids aren't. It is more noticeable in reading, where these kids get no phonics instruction, but the high SES kids eventually get IEPs and massive services to support terrible reading comprehension.

But, they are better off than being in Minneapolis public schools, where they would get no phonics and TERC investigations.

Plenty of low SES charters here are a total disaster. they may not be quite "guide on the side" but the teachers largely have no idea content matters. So there are no drills in math, no sense of what must be known year to year. No urgency.

Another big factor I see here is the "school expects home to teach math facts, but forgets to tell home that." A typical example for me is parents are shocked to find out their 4th grader is not competent at multiplication, and teacher is recommending summer school. They come to me to ask what is going wrong, and how do they help their child. Among other things, I suggest they ask teacher "how many minutes a day is spent on math facts in class?" They do, and receive the response "none".

Meanwhile, I see other schools where the parents are involved but to negative effect. In another typical example, the parents provide a steady steam of complaints if their child is not getting an A. This encourages group work and discovery learning, rather than tests that can be graded.

PALISADESK:

HAINISH: "Palisadek, if you are correct, then low-SES students in high-SES-area schools should be worse off than those in low-SES schools."

I think this may well be true, for several reasons. As Allison explained, the low-SES kids don't have the outside tutoring/afterschooling etc. that higher-income families routinely provide, and they tend (this is a generalization) to respond poorly to unstructured learning situations, which much "group work" and "exploratory learning" seems to be. They haven't got the resources at home or school to do artsy projects, may not have access to a computer or the Internet (or even a telephone!) at home, may have other responsibilities after school, not be able to afford field trips and school clubs/sports etc.

A previous school I worked at was in a neighborhood separated by a large city park from a very wealthy area of manicured million-dollar homes. The school for that neighborhood served these very affluent families, who comprised most of the enrollment, but on the edge of the neighborhood, bordering a freeway, there was a smallish public housing project. The children there also attended this school. So you had the very poor and the extremely rich. The school got allocated some extra special education staff for the "project" kids, but both socially and academically those children were isolated and tended to be academically unsuccessful. A top teacher from my school transferred there a few years ago and tells me that the great divide is still present, and the school does not have the kind of supports low-SES kids need.

For example, at my school the library has been kept open after school for parents and children to come in and use the computers for research, skill practice, homework and so on. Even though math facts are taught, many children need much more practice than can be given in class; we recommend some online sites for practice and pay for some sites where children can practice reading skills online (about 40% of our students have internet at home). Teachers also provide tutoring and support over the lunch hour and run academic clubs like math clubs and spelling clubs to reinforce basics in an engaging way.

Upper-income schools don't, in my experience, provide this kind of thing. Their students are leaving after school for Little League, swimming, horseback riding and gymnastics. Our students are leaving to care for younger siblings or help mom and dad at the bakery.

Adding to the difficulty is the fact that the lower-SES parents feel uncomfortable in a milieu of affluence (less so if it is a mix of working poor and working class), so parents aren't as involved in the school as they would be in one that was more reflective of their own social station.

One benefit, we do get away with a lot of direct teaching (phonics included) even though it is less than optimal. I compared my school's test results with those of one near my home, which has a median family income of 250K (I live on the poor side of the highway, LOL). My school roundly trounced this school, despite being 60% ESL and 95% nonwhite. Test results are only one indicator, but it does show that our kids are learning and we hope they will have a chance to make their way in the world.

the problem in a nutshell

Why does accelerating algebra for everyone not help? Moving on the same curve with a relabelled x axis does not change the curve.

Consequences of Common Core

Nationally, there is a movement toward standards-based teaching in mathematics education.

The main idea of standards-based teaching, that every school and classroom works from a common defined set of standards, and that therefore, a curriculum will  be designed and then implemented based on those standards, certainly sounds better than the alternative of no standards. Sadly, too many of our schools don't even know what standards are, and they define what they teach by how far through the textbook they happen to got.

Standards are enforced by assessments that claim to test exactly the standards. The whole thing is then an engineering process: write standards, write assessments to match standard, tell teachers to teach exactly that, repeat until proficient.

With the standards based zeitgeist also come the ideas of data driven instruction--check the outputs! test what they know! adapt accordingly. This gives you time to see if your student will pass the assessment before the student takes it.

But most schools still don't know how to teach to these standards, so they look to textbook publishers who claim to help. Because this push was national, Common Core was adopted nationally, giving the textbook publishers one non-moving target to hit.

The good news: Spiral Math is losing its grip. It would seem that publishers couldn't figure out how to map standards onto their spiral's scope and sequence, and opted instead to order their books to match Common Core. And schools want new textbooks. Because how can you do interim data driven instruction assessments in november and checkbox which standards your kids have passed if they haven't seen the material completely until May?

The bad news: at least some of the new textbooks teach Common core standards in order. exact lexicographic order in some cases. First the operations strand, then number and operation base ten, then  number and op fractions, then the decimals, geometry strand, etc.

Why is this bad? Because while Common Core is far more coherent than most, if not all state standards to date, the authors did not seem to understand that this line-by-line method would be how schools and publishers implement the teaching of the standards.

But, at least now it will be a lot easier to afterschool your child with Singapore's Primary Math because at least the sequence is stable.

More on cargo cult education in standards based mathematics teaching in a couple days.

Our schools do ruin Singapore Math

SteveH has said a few times "Our schools would ruin Singapore Math." Well, we can dispense with the "would". Schools routinely ruin Singapore Math, just as they ruin every other initiative brought in. Not all schools do, of course, but badly run schools ruin everything --professional development, curricula, assessment, leadership initiatives. Bringing in good materials to a bad school doesn't work at fixing the school. (Whether it saves an individual student here or there is an open question.)

I have now seen schools ruin Singapore math. They do this by not reading Primary Math's own materials, not understanding or seeking out anything about how Singapore taught using Primary Math, by not getting their staff any professional development at all for using Singapore Math. So their teachers know nothing about how the lessons are supposed to be taught, what "Concrete->Pictorial->Abstract" means, and teach math the same way they taught before. Typically, they don't even get them a full complement of materials. I've seen schools where the teacher NEVER read the teacher's guide, and just assumes that since the textbook looks simple to them, there's nothing more to it.

Then they ruin it some more by teaching it out of order to "match the state standards"(more on this in an upcoming post next week.) Since nearly everything in Primary Math is carefully predicated on what has been taught before, this makes no sense at all.

I've now seen schools ruin Lemov's Teach Like a Champion, too. They do this by using it as a micromanagement tool--lesson plans must look exactly like this, word choice must sound exactly like this,   etc. to beat the teachers into submission. Then they ruin it some more by using the lack of techniques present during a class to penalize a teacher during an evaluation.  Nothing like making the teachers hate the administration, or that their lesson planning is a waste, or that all "teacher improvement" is really just a cover for a method for trying to fire a teacher to make the school function!

I've now seen schools ruin PLCs, too.

What is the common denominator?  cargo cult education.  To shamelessly quote myself:

Unless they understand what's underneath the "lessons of the high performing school" (the high performing parents, the high performing teachers, the high performing students) then it won't matter. Unless the "lessons" they grab are that they need teachers who already know classroom management skills and content, need solid curricula that can be built to mastery, need ability grouping rather than differentiated instruction, need schools that already enforce discipline and control their students' behavior, need raised expectations for all students, and more, then they will be missing something essential."

In each case, these schools have no idea what the actual function of teaching is, what knowledge you're supposed to impart to the students. So they grab PLCS, or TLAC, or Singapore Math, but it's just coconut shell headphones. Their teachers don't know that their students don't know 10/9 is a fraction; their teachers don't know that you can't teach the chapters of a math book in another order; their teachers don't know why place value works. Their principals don't know that their teachers haven't read the Teacher's Guide at all; their principals don't know that "accountable talk" isn't the same thing as explaining your bar model; their principals don't know that their teachers didn't follow any scope and sequence this year; their principals don't know what their teachers and students don't know. And they may test prep their way to better scores, but their students will still not know enough to succeed in high school, let alone college.

The real question is: can you find a school that isn't practicing cargo cult education? How would you know?

Does your student know 10/9 is a fraction?

American elementary and middle school mathematics programs are poor in myriad ways. They lack breadth and depth. They lack reasoning. They lack precision. An object lesson in all of the above is the common use of analogy to teach math, even in grades 5-8.

 Math by analogy is when teachers substitute ideas completely unrelated to math in order to make some concept "easier". Usually, this is because they themselves do not understand the meaning behind what they are teaching, so they cannot explain it accurately. Math by analogy substitutes presumed common context for reasoning. Yet most young students don't share enough common context to build the analogous connection anyway, even if they can abstract away from the literal -- something most children cannot do. And if you are are ELL, it is probably entirely worthless. Plus, the analogy is by definition imprecise, so its correctness will break down with even the slightest scrutiny.

 You see math by analogy in both big and little examples, from the use of it to "explain" greater than and less than to its use in teaching place value. The most common analogy I see used by teachers and their books is that "a fraction is part of a whole".This analogy has devastating results. I routinely (in 100% of classrooms not using Singapore math, in more than 50% of the students) hear:

•  1. "there's no such thing as ten ninths." that's the majority response in classroom after classroom. Why? Because a fraction is PART of a whole. How can a part of a whole be bigger than the whole? What's the whole then?  
• 1b. therefore, they believe no fraction can be bigger than 1.
• 2. "You can't divide 6 things among 7 people." 6 things isn't one whole. It's 6. 
• 3. "three thirds is A Whole." Not one. 
•  3b. Therefore, they don't know 3 divided by 3, written as a fraction, is 1. I often hear of students who ask "is this a division problem or a fraction problem?" 

 Additionally they don't know decimals are fractions. how could 1.2 be a fraction? Twelve tenths isn't a fraction, remember?

 These problems are so severe because these students have teachers who manage not to notice these errors. No problems in their books, no lesson script in the teachers guides illuminates this to the teacher. They only see the most trivial of problems. 10/9 is beyond the pale.

 The correct explanation is that a fraction is a number. What number? A number defined on the number line as follows:

1/3 is the point on the number line when you break the unit length into 3 equal length parts, and take 1 part. the endpoint of that part is 1/3.

4/3 is the point on the number line when you break each unit length into 3 equal length parts, and take 4 parts. the endpoint of those parts is 4/3.

 Yes, teachers will need to build up to this. They should do so.

Allison on Khan-love in Minnesota

I'm seeing the same thing as Catherine: schools love khan.

Khan isn't going to teach k-8 kids anything, but schools love it anyway.

a) now they think they don't have to spend money on textbooks for elementary kids (first hand have heard that directly from a curriculum director)

b) now their teachers don't need to know how to do the math, khan will do it for them. (heard that directly from an instructional coach who champions Teach Like a Champion)

I know 2 other elementary teachers who love it because now they can do fun Terc things.

I only know one person who is anti Khan here in the establishment. Her very sane complaint: the man teaches completing the square without even drawing a square. It's nothing but computational and procedural fluency for him. It's the opposite of actual instruction, but now schools will use it and instruct even less.

I'm surprised nobody said one of Khan’s most significant achievements is that he has enormously expanded the world’s access to a master teacher.

Does anyone want a STEM career anymore?

As the self-described 99% show the country what a wasteland a liberal arts education is, the current administration says STEM careers will transform (or is it save) America. But there's been a number of articles in the last few days about why American students today aren't choosing STEM careers.
in today's WSJ, Generation Jobless: Students Pick Easier Majors Despite Less Pay, the article opens with this:

Biyan Zhou wanted to major in engineering. Her mother and her academic adviser also wanted her to major in it, given the apparent career opportunities for engineers in a tough job market. But during her sophomore year at Carnegie Mellon University, Ms. Zhou switched her major from electrical and computer engineering to a double major in psychology and policy management. Workers who majored in psychology have median earnings that are $38,000 below those of computer engineering majors, according to an analysis of U.S. Census data by Georgetown University.

"My ability level was just not there," says Ms. Zhou of her decision. She now plans to look for jobs in public relations or human resources.

The NYT had this article, "Why Science Majors Change Their Minds (It’s Just So Darn Hard)" (pointed out by Glen a few days ago.)
it states:

Studies have found that roughly 40 percent of students planning engineering and science majors end up switching to other subjects or failing to get any degree. That increases to as much as 60 percent when pre-medical students, who typically have the strongest SAT scores and high school science preparation, are included, according to new data from the University of California at Los Angeles. That is twice the combined attrition rate of all other majors.

Why the attrition? Some are the usual suspects: from the WSJ piece again:

For 22-year-old Ms. Zhou, from Miami, the last straw was a project for one of her second-year courses that kept her and her partner in the lab well past midnight for several days. Their task was to program a soda machine. Though she and her partner managed to make it dispense the right items, they couldn't get it to give the correct change.

Such unpreparedness in part explains this (from NYT piece): "Ben Ost, a doctoral student at Cornell, found in a similar study that STEM students are both “pulled away” by high grades in their courses in other fields and “pushed out” by lower grades in their majors." But so does the burnout factor from the death march through calculus, as illustrated by "MATTHEW MONIZ bailed out of engineering at Notre Dame in the fall of his sophomore year...He had scored an 800 in math on the SAT and in the 700s in both reading and writing. He also had taken Calculus BC and five other Advanced Placement courses at a prep school in Washington, D.C., and had long planned to major in engineering....But as Mr. Moniz sat in his mechanics class in 2009, he realized he had already had enough. “I was trying to memorize equations, and engineering’s all about the application, which they really didn’t teach too well,” he says. “It was just like, ‘Do these practice problems, then you’re on your own.’

They quote Mitchell J. Chang, an education professor at U.C.L.A. who says it isn't just weak K-12 prep that causes this washout.

"You’d like to think that since these institutions are getting the best students, the students who go there would have the best chances to succeed,” he says. “But if you take two students who have the same high school grade-point average and SAT scores, and you put one in a highly selective school like Berkeley and the other in a school with lower average scores like Cal State, that Berkeley student is at least 13 percent less likely than the one at Cal State to finish a STEM degree.”

His argument seems to be that the kids at Cal are better prepared than the kids at CSU, so more of them should succeed, if the issue was really k-12 prep. I don't think that gets to the heart of the prep matter though. The kids at Cal, Notre Dame and the like are Used to Succeeding, and they aren't succeeding. This is a huge blow to them, at the same time that the intro courses are often seen as the drudgework to get to the electives, a point made in the NYT article. It feels better to get As in psych than B-s in EE.

Continuing,

Some new students do not have a good feel for how deeply technical engineering is. Other bright students may have breezed through high school without developing disciplined habits. By contrast, students in China and India focus relentlessly on math and science from an early age.

“We’re in a worldwide competition, and we’ve got to retain as many of our students as we can,” Dean Kilpatrick says. “But we’re not doing kids a favor if we’re not teaching them good life and study skills.

So many fall off. And what about the ones who make it?

You work harder for lower grades than your peers, and the payoff is either a) a career path where your employer is constantly lobbying the govt to drive down your pay by increasing immigration, or b) a career path where you front load all of your risk onto a low probability lottery ticket to the world of academia just as the higher ed bubble is bursting.

Not depressing enough? Read Arnold Kling's "What If Middle-Class Jobs Disappear?" Kling suggests high unemployment now is structural, coming from a new phase of an economic transition away from plentiful high paying white collar jobs, just as prior restructuring moved away from plentiful high paying blue collar jobs.

Using the latest Census Bureau data, Matthew Slaughter found that from 2000 to 2010 the real earnings of college graduates (with no advanced degree) fell by more in percentage terms than the earnings of high school graduates. In fact, over this period the only education category to show an increase in earnings was those with advanced degrees.
The outlook for mid-skill jobs would not appear to be bright. Communication technology and computer intelligence continue to improve, putting more occupations at risk.

For example, many people earn a living as drivers, including trucks and taxicabs. However, the age of driver-less vehicles appears to be moving closer.

Another example is in the field of education. In the fall of 2011, an experiment with an online course in artificial intelligence conducted by two Stanford professors drew tens of thousands of registrants. This increases the student-teacher ratio by a factor of close to a thousand. Imagine the number of teaching jobs that might be eliminated if this could be done for calculus, economics, chemistry, and so on.

So much for that lottery ticket to academia...but what about the private sector? Kling says:

...the main work consists of destroying someone else's job. Garett Jones has pointed out that the typical worker today does not produce widgets but instead builds organizational capital. The problem is that building organizational capital in one company serves to depreciate the organizational capital somewhere else. Blockbuster video adversely affected the capital of movie theaters, Netflix adversely affected the capital of Blockbuster, and the combination of faster Internet speeds and tablet devices may depreciate the organizational capital of Netflix.

The second challenge is the nature of the emerging skills mismatch. People who are self-directed and cognitively capable can keep adding to their advantages. People who lack those traits cannot simply be exhorted into obtaining them. The new jobs that emerge may not produce a middle class. Instead, if the trend documented by Autor for the period 1999-2007 were to continue, most of the new jobs would be low-end service jobs, for which competition will tend to keep wages low.

He goes on to posit some possible futures. He's not optimistic.

update: Kling link fixed. Thanks, ChemProf!

Bonnie on peer review and textbooks in computer science

Bonnie writes:

Peer review in computer science is very weird, because unlike other fields, we mainly publish in conferences rather than journals. It is a huge issue at tenure time because tenure committees assume that conference publications are meaningless - and they are in most fields - but not in computer science. Most of our conferences have acceptance rates of around 25%, and the top conferences are below 10%. That is where the peer review is happening in our field.

But that is research peer review, and I thought we were talking about textbooks. In CS, there are textbooks for certain "standard" courses - intro to programming, databases, operating systems, theory of computation, and a few others. The only real churn that I see is in the intro to programming area, because every time a new programming language hits the scene, you get 10 new books doing CS 1 and 2 in that langauge. In databases and operating systems, the same 3 or 4 authors have dominated for 20 years, issuing edition after edition of their book. So in reality, professors adopt the textbook, which isn't purchased or read by most of the class anyway, and then add lots of their own material. For courses that have no reasonable textbook - for example, everyone is adding courses on Android programming right now - we simply have the students buy a book aimed at professional developers, or cobble together notes on our own.

And here is Allison on peer review in mathematics.

the real world

Speaking of ditching the daily lesson plan, Allison wrote:

For those so enamored with preparing kids for the real world, why do they want school at all? Wouldn't child labor better? That would integrate everything.

LA Times: Excellent and terrible teaching found in the data

The LA Times is beginning another series of articles about LAUSD, this series based on access they've had to LAUSD's longitudinal test data. Prior articles focused on money, with the Times creating an easily readable database listing all LAUSD employees' salaries. This time, they are focused on teaching.

The first article's intro says "A Times analysis, using data largely ignored by LAUSD, looks at which educators help students learn, and which hold them back."

To accomplish this,

The Times obtained seven years of math and English test scores from the Los Angeles Unified School District and used the information to estimate the effectiveness of L.A. teachers — something the district could do but has not.

The Times used a statistical approach known as value-added analysis, which rates teachers based on their students' progress on standardized tests from year to year. Each student's performance is compared with his or her own in past years, which largely controls for outside influences often blamed for academic failure: poverty, prior learning and other factors.

The article profiles a couple of strong and weak teachers, and apparently, more articles are forthcoming that will do more profiling. It seems that after analyzing the data, the authors went to the classrooms of those in the top decile and bottom decile for student improvement to view the teachers in action.

Miguel Aguilar at Broadous Elementary School is one of the strongest. "On average, his students started the year in the 34th percentile in math compared with all other district fifth-graders. They finished in the 61st."

That's an impressive improvement. I wish I understood enough details of the underlying scoring to know how this relates in standard deviations. Is improving a student one standard deviation when they one below the mean as difficult as improving a student one standard deviation when they are at the mean? Certainly it's not the same effort to move a students from 1 standard dev away from the mean to 2. What assumptions can be made about equal difficulty in movement of scores measured in percentile?

The article repeats what we all know as well: you must raise the bar.

"On visits to the classrooms of more than 50 elementary school teachers in Los Angeles, Times reporters found that the most effective instructors differed widely in style and personality. Perhaps not surprisingly, they shared a tendency to be strict, maintain high standards and encourage critical thinking.

But the surest sign of a teacher's effectiveness was the engagement of his or her students — something that often was obvious from the expressions on their faces."

The article goes on to argue that their analysis shows that excellence in teaching and weakness in teaching matter a great deal.

"Among the findings:

• Highly effective teachers routinely propel students from below grade level to advanced in a single year. There is a substantial gap at year's end between students whose teachers were in the top 10% in effectiveness and the bottom 10%. The fortunate students ranked 17 percentile points higher in English and 25 points higher in math."

The LAT is creating a database for release "in the coming months." I can't wait. I wonder if it will shed light on the value of mediocre teaching?

It's great to see this article. With luck, it will propel other journalists to perform similar studies. Perhaps some enterprising ed bloggers can FOI this information for their district, and perform the same analysis.

msmi 2010 Followup

msmi 2010: Institute on Fractions is now history. YAY!

I'll leave Catherine and any others to talk about their take on it. I'll have some future posts on what I think are the biggest lessons parents and teachers need to help their students understand fractions, but this is just a roundup.

In the course of 5 days, we covered: definition of a fraction, equivalent fractions, decimals, addition, subtraction, multiplication, division, decimals again, and percent. We had more to do, but we couldn't get to it.

Based on the reaction of the teachers, it was a success. I have NEVER had a class of students where so many students worked so hard. No matter what their background, everyone tried to do the problems. Their effort meant that as the week went on, the students were more engaged and more knowledgeable. The teachers also built up their camaraderie with each other.

Based on their personal comments to me and the anonymous survey I gave at the end, their overall impression was quite high. Several teachers told me that this course was the first time anyone had ever explained how to think about fractions. One told me it was "a revelation" to them, another told me this was the first time they'd see a way to visualize multiplication of fractions. Most responded to our survey saying that this material had changed how they would teach permanently. Several teachers had a different kind of revelation, too: that other people in other schools/cities/states knew and felt as they did. They were connecting the dots not only on fractions, but on the state of math education.

Not that everything was perfect. I was terribly out of practice for being a teacher--bad board technique, bad handwriting, bad short hand in my own thoughts and words, instead of being clear, specific and slow.

I made several errors in sizing up my audience too. I assumed that since I had told the principals what to expect, that they had told their teachers. I assumed that teachers, given a pointer to a web site that had, e.g. Wu's CV on it, would have read such.

The biggest complaint was that it was too much material/days too long, and not enough worked out examples. One solution to the latter is to strongly encourage the teachers to read the textbook a day ahead of time. But part of that is the nature of the beast: there is an enormous deficit of knowledge to overcome. Elementary math teachers didn't go into that field because of their stength in fractions. The breadth of math inexperience-experience even for teachers of the same grade was very large, yet being math experienced didn't quite help, because while those teachers probably followed Wu's proofs more easily, applying his ideas to actual math problems was still a new universe to them, and their skill was sometimes a hindrance, because he was asking them to think an entirely different way than they were used to.

Lastly, I'm thrilled to have met all the people involved. Wu is a delight to work with/for, and I'd do this again with him wherever we can. He was personable and charming as well as brilliant. His wife was just as delightful. CassyT, KTMer, is an exemplary woman. She's a brilliant teacher and student of human nature, and her insights into teachers saved me countless mistakes. She shared her expertise with me in countless ways, and the whole thing would have fallen apart if not for her.

So, where to go from here? First, more Wu institutes! Let's bring MSMI to your locale! Second, the really big thing is to help teachers turn what they learned here into changes in their school. That's no small undertaking. I'll talk about that more in the next post. Last, more documents for everyone: condensing of Wu for parents and teachers. I'm sure you'll see work product of that around here shortly...

Allison explains absolute value inequalities

Catherine: also, I have completely forgotten how to set up and solve a simple inequality involving absolute value

Allison: You need to think about what it means.

The absolute value of any number equal to or greater than 0 is itself. x-->x

The abs value of any number less than 0 is the number*-1, or the number with its sign dropped: x --> -x.

So break down the inequality you see into other inequalities.

|x| - 2 > 3

means
|x| > 5:

allowable positive values of x satisfy
x > 5

allowable negative values of x satisfy
-x > 5

you "solve" this by switching sides (do you know why you're allowed to do that?)
and that becomes
x < -5

Now, you do |x - 3| < 5 in a similar way. To make yourself less confused, write (x-3) = y
and work on
|y| < 5

Then after you've got that into equations without the abs value, sub back in the x-3.

Targeting Differentiated Instruction

In the 2 years of participating in KTM, I have come to the conclusion that differentiated instruction is gravest of all of the myriad problems in K-12 today.

In a world where Paul B has one classroom of 7th graders whose current proximal development covers a 9 YEAR spread (from 3rd to 11th grade), where PalisadesK says her secondary school is in crisis because numbers of kids who are entering 9th grade don't even register as having 4th grade math, it seems clear that nothing can be fixed without grouping by ZPD/ability/whatever you want to call it.

Parents do not know that this is happening. They have no idea that all of that lip service about "teaching to students with stages of development in different learning styles" is meant to paper over 9 YEAR disparities, that children could be entering high school already 6 YEARS behind.

If parents--no, if people-- did know, I believe they would be in uproar. This may be the only place where the bulk of parents and nonparents would agree.

So how can we crack this nut? It isn't top down; there is no chance of changing the essentially mandated differentiated instruction from the top of the ed school chain, or from the top of the superintendent's chains in nearly any districts. There are many many forces leading to differentiated instruction in order to create "inclusion", "diversity" and a whole bunch of other feel-good social goals from the top. The bigger powers that be, politicos at the state level and the like probably also have no idea how much of crisis this is, because again, they don't see what's happening in an individual classroom. They know only about the means and medians of achievement gap; they don't know that that achievement gap translates into more than half a decade of dispersion in a 7th grade classroom.

The only chance that I see is to crack it open by going directly to the people.

A documentary that interviewed actual middle school teachers, who would speak honestly about the disparity of skill/ability/proximal development in their individual rooms (NOT at the school level, or district level, but IN THEIR ROOMS), that showed the desperation of good teachers trying to teach curricula across a 3, 5, 7, year gap, might be huge.

Especially if the documentary could show it across the country: a national problem in EVERY STATE. Schools where 20k per pupil spending by the district and those with 8k, etc.

The documentary should be done by some creative, hip, young film maker, not a policy wonk, who could get teachers telling the story themselves, not using statistics or analysis of data, viewable on the web, or in short bursts (or given to Andrew Breitbart.)

I don't mean to imply that ending differentiated instruction will suddenly bring 7th graders 5 years up in a year. But none of those problems can possibly be solved inside the group delusion that is the differentiated instruction classroom. Yes, bad curricula will still need fixing. But you can't make Singapore Math 7A work in Paul's classroom, either.

Well, what do you all think?

Always worse than you think, Congressional version

It's always worse than you think. CPSIA, that is. The Consumer Product Safety Improvement Act, passed overwhelmingly by Congress, is doing damage to vast swaths of childrens' fashion and toy industries as it mandates, among other things, outrageous new testing and tracking by manufacturers, retailers, and resellers to prove products are in line with overly stringent lower limits on lead levels, phthlates, and other chemicals.

The law is so poorly written that it's affecting non-childrens' products in all sorts of industries. Of course, it's mostly small businesses that can't afford to come into compliance. The educational products industry is filled with such small vendors. Here's a quote from one in an AP story:

But some small businesses, like American Educational Products in Fort Collins, Colo. — it sells classroom teaching aids like flash cards, animal models, globes and relief maps — say the testing and labeling costs are crippling to their operations even though their products are safe. They want the law amended to exempt products that present little or no risk to young children.

"The challenge as a small business is that I cannot do it all (the testing) immediately," said Michael Warring, president of AMEP. "I would have to spend a full year of revenue to test every product I sell."

Warring recently laid off four of his 70 employees. In his 15 years with AMEP, he has not had one safety recall or complaint about lead.

Even so, Warring says he is required to test samplings of all products he makes and sells for young children, which he said costs about $2,000 per product. The tracking labels will add another cost, he says, since they must be a permanent marking on each product.

Another industry hit hard is the scientific equipment industry. Here are some stories as they affect science classrooms.

From the Amend The CPSIA blog,

Heathrow designs and manufactures items for use by trained laboratory technicians. ... Heathrow directly employs 13 individuals in Vernon Hills, Illinois and 1 in Great Britain. Heathrow recently received a request from one of its U.S. customers to certify that its products meet the standards set forth in the CPSIA.

Why, you may ask, would a company that designs, and manufactures, products for use by trained laboratory technicians, in professional labs, be asked to certify that its products meet standards set forth in a law that deals with safety standards for children’s products? The answer is that..this particular customer of Heathrow sells the Heathrow product range into the middle school science classroom marketplace...Therefore, they think they need to have on file certification from their suppliers that these products meet the CPSIA standards...Our products are not designed for use by children... if products are not designed for use by children, they are not subject to the CPSIA. However, many companies are spooked by the fact that this law has mandatory $100,000 per occurrence fines and felony criminal sanctions. They do not want to go to jail for selling products that violate the CPSIA, nor can they afford to risk $100,000 per occurrence fines.

So, they will either get their certifications or drop the products. This means that our products will no longer be available for use by middle school science teachers (who apparently found a use for them in teaching biology, chemistry and other sciences)

This isn't the only manufacturer that won't be passing CPSIA test for its microscopes. The solder on microscope light bulbs fails the lead test too. So
no microscopes at all.

But that's okay, you probably wouldn't have had anything to look at anyway.
From here:

"First, Michael Warring of American Educational Products reports that a school opted to stop using AmEP's rocks to teach Earth Science and will instead rely on a POSTER... The continued ragging of consumer groups about "toxic toys" sullies the reputation of all good companies and their good products. In this case, rocks take on the "toxic" tag because they contain uncontrollable amounts of base elements found in nature.

It gets worse. Nearly all science kits could fall because of the lead in the insulation on the wires, as they did in the case of the Potato Clock.

From the above blog again:

"recently a manufacturer of the Potato Clock decided to test its version for compliance with the newfangled CPSIA. In their eager beaver-ness, they shot themselves in the foot, discovering (horrors) that the insulation on the product's potato wires contain trace amounts of lead over the arbitrary limits of CPSIA...

First, the company decided that since it now knew of the test failure, it had an immediate reporting obligation under CPSIA Section 15(b). In addition, they concluded they had an obligation to immediately stop sale, since continuing to sell would be another "knowing" violation - yes, kids, that's a felony with possible penalties of jail time and asset forfeiture (goodbye house and car!)...

The CPSC, apparently, upon receiving this (unwanted) 15(b) report concurred - yep, the wire insulation exceeds the standard, and yep, you have to stop sale. No recall was required by the CPSC BUT the company appears to have decided almost immediately that an informal recall was mandated. Why might they have decided such a thing? Well, perhaps they had a generalized fear of liability from dealers who might be sued for selling this "dangerous" device if it ever came to light that the product had impermissible lead in the wire insulation....

But the WORST part of this story, the most chilling, is the part about the wire insulation. The Potato Clock was recalled for having too much lead in the wire insulation. Why did it have lead in it at all? Wire insulation contains lead because it is recycled vinyl, probably recovered principally from scrap of other wire...

The real problem comes from the fact that the Potato Clock utilizes "ordinary" wire. Everyone and everything utilizes "ordinary" wire. No specially-coated wire is used in children's products and even if it were available, it would be too expensive for this kind of application. Potato Clocks should use "ordinary" wire. If ordinary wire will always fail the CPSIA standards because of its insulation, then everything using wire in schools can't be sold for use by children under 13 years of age. This means, among other things, no electricity education before the 7th grade in this country (and only for the 13 year olds in the room - the 12 year olds will have to leave the room until their birthday)."

On the bright side, at least it will end discovery learning.

Physics Education Continued

In a prior post, I discussed the problem of college students having poor physical intuition both before and after taking university physics. In another post I will discuss David Hestenes' proposed solution to this problem, but first I wanted to provide some examples of what kinds of errors these students are making.

Hestenes et. al. developed a test called the Force Concept Inventory, but they've embargoed online versions of it (it's available to you if you can prove you are a physics teacher or professor.) The following questions are similar to questions on the FCI, but I've respected their embargo, and adapted them from similar questions. Their own questions are taken from other papers as well, as the literature is filled with examples of how physics students don't understand basic mechanics.

Here are two test questions, the first adapted from Students' preconceptions in introductory mechanics, J. Clement, Am. J. Phys. 50(1), Jan. 1982, and the second adapated from Rule-governed approaches to physics--Newton's third Law, D. P. Maloney, Phys. Educ., Vol 19, 1984. Note that my adaptations haven't been tested on thousands, so they may not be as crystal clear as I hope...

1. A ball is tossed from point A straight up into the air and caught at point E. It reaches its maximum height at point C, and points B and D are at the same height above the ground. IGNORE AIR RESISTANCE.
Try to imagine that "up" on the page is the z direction, and that the horizontal direction is x. No motion is occurring in x.




a. Draw with one or more arrows showing the direction of each force acting on the ball when it is at point B.
b. Is the speed of the ball at point B greater, lesser, or the same as at point A?
c. Is the speed of the ball at point D greater, lesser, or the same as at point B?


2. Consider the following diagrams of two blocks on a frictionless surface and answer the following questions. Ignore air resistance.
a. Assuming both blocks are at rest:


How does the force that A exerts on B compare to the force B exerts on A, if A and B are equal in mass?
How does the force that A exerts on B compare to the force B exerts on A, if A and B have different masses?

b. Assuming both blocks are moving to the right with velocity v:


How does the force that A exerts on B compare to the force B exerts on A, if A and B are equal in mass?
How does the force that A exerts on B compare to the force B exerts on A, if A and B have different masses?

c. Assuming both blocks are moving to the left with constant acceleration a:


How does the force that A exerts on B compare to the force B exerts on A, if A and B are equal in mass?
How does the force that A exerts on B compare to the force B exerts on A, if A and B have different masses?
---
While the actual test employed some randomization and various other elements (set values for the masses, e.g.) the results for this last question were that less than 10% of experienced students (those who had taken college physics) got the right answer using the right reasoning, and 0% of the novice students (those who had not yet taken college physics) got the right answer.

UPDATE: See, I told you I hadn't vetted the questions. Updates are above in BOLD. College Physics above means college students taking a standard first term mechanics course. In the test they did with question c, the students were junior or senior year chemistry students who were required to take a 1 year physics course as a prereq for their major. The author was at Creighton University, so presumably these students were at Creighton University as well. The author points out that at least half a dozen of these students who got these wrong had also taken the MCAT, and possibly had studied physics AGAIN as well.

SECOND UPDATE:

how about some answers?

Problem 1: a. There's one force on the ball. it's Gravity, pointed down. b. The speed of the ball at B is less than the speed at A. The speed drops continuously until we reach C, in fact. c. The speed of the ball at B is the same as at D. In fact, the speed of the ball at any height X above the initial A is the same whether going up or going down. The ball speed depends only on height above our origin.

Problem 2: the answer to all problems is the same: the force exerted by A on B is the same as the forced exerted by B on A.


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

Fixing Physics Education: Modeling Instruction

In prior posts, I referred to work done by David Hestenes and his colleagues at Arizona State University addressing the dismal results of traditional university level physics instruction. Hestenes and others developed the Force Concept Inventory, FCI, to demonstrate that even after a year of traditional instruction, college students had failed to create proper models in their own minds for how mechanics actually works. Specifically,

"Before physics instruction, students hold naive beliefs about mechanics which are incompatible with Newtonian concepts in most respects.
• Such beliefs are a major determinant of student performance in introductory physics.
• Traditional (lecture-demonstration) physics instruction induces only a small change in the beliefs. This result is largely independent of the instructor’s knowledge, experience and teaching style. "

Hestenes has gone forward from there, and developed a new curriculum design for high school and college physics instruction, and has pushed this curriculum design out to high school and college physics teachers on his own, and more recently, through NSF backing. He calls this new type of instruction modeling instruction.

What is meant by modeling instruction? He means that you learn physics by constructing appropriate models for the interactions in your system, and then you apply inference to your model to solve whatever problem you have. The emphasis is on getting the student to recognize the model at hand by getting them familiar with constructing models in the first place. What's a model? A model is a representation of your system and its properties. A model tells you everything you need to know. So a model tells you your system, the boundaries of your system, the state variables inside your system, the initial conditions of your system, the transition function for the state variables in that system, and whatever interactions you need to know. This sounds vague, but the point of the model is that you can explicitly say whether or not you've got everything you need to infer what happens if you write it all down.

In modeling instruction, the idea is that "the modeling method approaches the problem of restructuring students’ intuitions by engaging them in explicit construction and manipulation of externally structured representations. In the case of mechanics (6), we have found it advisable to engage students in explicit comparisons of the three major misconceptions in Box 1 with their Newtonian alternatives. When these three are adequately treated, many other misconceptions about mechanics fall away with them. "

To facilitate the teaching of mechanics by modeling instruction, the modeling group at ASU created course materials and curricula for a high school level mechanics class. Their modeling method for mechanics explicitly teaches 10 models, five of them models of motion (kinematical models): constant velocity, constant acceleration, the simple harmonic oscillator, uniform circular motion, and a collision model; and five models of force (causal models): the free particle, the constant force, the central force, the linear binding force, and the impulsive force. The idea is that by explicitly organizing the ideas of motion and force in this way, the student will see the common physics in each. This is as opposed to organizing ideas around "problems".

Here's an example. "oh, that's a projectile problem", " oh that's a block-siding problem" "oh, that's a orbit problem" is a typical way a student might think about the physics problem in front of them, but it doesn't help elucidate what were the relevant features of the problem at hand, whereas recognizing "oh, that's a constant velocity problem", "oh that's a free particle problem," "oh, that's a central force problem" leads you immediately to know or be able to infer the geometric structure, the interaction structure, the force structure, the changes over time, etc.

Enough talk! Let's jump in. Here are the course notes for unit on the free-particle model. The instruction goals are to use the free particle model to develop intuition for Newton's First Law (commonly stated as "an object in motion tends to stay in motion; an object at rest tends to stay at rest"), for Newton's Third Law , and to correctly be able to represent forces as vectors.


1. Newton’s 1st law (Galileo’s thought experiment)


Develop notion that a force is required to change velocity, not to produce motion
Constant velocity does not require an explanation.

2. Force concept

View force as an interaction between and agent and an object
Choose system to include objects, not agents
Express Newton’s 3rd law in terms of paired forces (agent-object notation)


3. Force diagrams

Correctly represent forces as vectors originating on object (point particle)
Use the superposition principle to show that the net force is the vector sum of the forces



4. Statics

•F = 0 produces same effect as no force acting on object decomposition of vectors into components

Continuing, the teacher's notes state "It is essential that you get students to see that the constant velocity condition does not require an explanation; that changes in velocity require an interaction between an agent and an object. We quantify this interaction by the concept of force. After the dry ice and normal force demos, one can use worksheet 1 as an opportunity to deploy the force concept in a qualitative way. It is important to carefully treat how to go about drawing force diagrams in which one represents the object as a point particle. Drawing the dotted lines around the object helps students distinguish between the object and the agent(s). "
And

"Newton’s Third Law ...Researchers have identified and categorized many such misconceptions, but two of them are particularly important, because they are persistent common sense alternatives to Newton’s Laws. Ignoring variations and nuances, these misconceptions can be formulated as intuitive principles.
I. The Impetus Principle: Force is an inherent or acquired property of objects that make them move.
II. The Dominance Principle: In an interaction between two objects, the larger or more active object exerts the greater force."

The notes then describe detailed demos and labs, with pre and post discussion points as well:

"It is an indirect goal of this activity to provide students an opportunity for arguing that a free particle, i.e. one subject to zero net force, will have a constant velocity. Also, students should conclude that any apparent change in velocity of an object indicates that a non-zero net force is acting upon it, provided that the observer is in an inertial frame of reference. ,,,
Make the point that when no force acts on the block in the horizontal direction, the block maintains constant velocity.
* Point out that an impulse applied perpendicular to the original trajectory does not result in the block making a right angle turn.
* Be sure to ask why they think the block continues to move once it leaves the hand. Some are likely to answer " due to the force of the hand."

The notes continue in this fashion, with specific notes on what demos/labs, how to guide them, what the appropriate leading questions are, when to ask for students' input, when to build to consensus.

This might seem like a fairly normal course, being taught with a normal lab. But the structure is different. More on that in the next post.


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

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

Mastery and Conceptual Understanding in Physics

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.

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 he doesn't say. Error correction in intuition can only occur and stick if the mastery of the manipulation of the equations is so strong that you 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

Not As Good As You Think

Catherine often writes that "it's worse than you think."

The Pacific Research Institute has a similar idea in their heads.
They have released a new movie, "Not As Good As You Think, the Myth of the Middle Class School"

From their website:

Not as Good as You Think: The Myth of the Middle Class School is a documentary that shatters the myth that “good” schools located in “nice” neighborhoods are shielded from the education crisis that pervades schools in poor, urban areas. Using available data on school performance and interviews with parents, students, principals, and school reformers, Not as Good as You Think confirms every parent’s silent fear: that their financial sacrifice and investment in an expensive home in a “good” school district is not yielding the achievement results needed to get their kids in good colleges and good jobs.

Based on the groundbreaking book Not as Good as You Think: Why the Middle Class Needs School Choice released by the Pacific Research Institute in 2007, the film features Lance Izumi, one of the nation’s foremost education scholars and reformers. Not as Good as You Think takes audiences on a tour of America’s best neighborhoods—from the posh areas of Orange County, California, to the hotbed of innovation, Silicon Valley, to the lush green hills of Tennessee—to reveal that schools in America’s middle class and affluent neighborhoods are not adequately preparing kids for higher education, or even operating under widespread corruption. The documentary also explores freedom and choice in public schools by venturing to Sweden, a socially progressive country that has successfully established school choice for all children, no matter the family’s income.

This is a different film than The Cartel, which Catherine posted about here.

You can buy the DVD for $19.95 from the above website.

Maybe we should all try to help set up a screening of the two films? A double feature? I'll bring the junior mints.