kitchen table math, the sequel: if you can't do it, you don't understand it

## Wednesday, December 22, 2010

### if you can't do it, you don't understand it

cross-posted at the Irvington Parents Forum:
Reviewing his son’s grade-school homework in the early 2000s diverted W. Stephen Wilson from his research in algebraic topology to question basic math education. At two well-regarded private schools, Wilson’s son had encountered the most widely used elementary math curricula, Investigations and Everyday Mathematics. Both encourage the use of calculators for multiplication and division, in line with a 1989 report from the National Council of Teachers of Mathematics that downplayed teaching arithmetic with pencil and paper. “What the schools were doing with math was beyond my imagination,” Wilson says. “I knew that my kid’s third-grade math wasn’t going to prepare him for college.”

A professor of mathematics in the Krieger School, Wilson teaches calculus to undergraduates. In 2006, he decided to conduct an experiment with his Johns Hopkins students. His Calculus I for the Biological and Social Sciences class that year bore close resemblance to the 1989 class. Their scores on the SAT math exam were nearly identical, and the two groups contained the same percentage of freshmen. Curious to see how they’d compare on the same exam, he gave the 2006 students the same 77-point final that he’d given the 1989 class. The results, he believes, confirmed his hunch that students were coming out of K–12 schooling less prepared for college math. When he compared scores by the grading scale in use in 1989, 27 percent of the 1989 students received As on the exam and 37 percent scored Bs. Only 6 percent of his 2006 students would have received As, 26 percent Bs.

[snip]

As another experiment, Wilson gave a short test of basic math skills at the start of his Calculus III class in 2007. The results predicted how students later fared on the final exam. Those who could use pencil and paper to do basic multiplication and long division at the beginning of the semester scored better on the final Calc III material. His most startling finding was that 33 out of 236 advanced students didn’t even know how to begin a long division problem.

Wilson says he wouldn’t be so against calculator use if teachers still taught multiplication and division by hand as well, regardless of the fact that few will ever do math that way as adults. “The theory that people should only learn what they are going to use as adults doesn’t make a lot of sense,” he explains. “If you take that to the extreme, there wouldn’t be much left to K–12 education. If someone is going to fly an airplane when they grow up, should we skip all the intermediate steps and just teach them how to fly an airplane when they are 10?”

[snip]

In 2006 he served as senior adviser for mathematics in the U.S. Department of Education, where he helped form the National Mathematics Advisory Panel. Since his return to his Hopkins classes in 2007, Wilson has continued to review K–12 curricula for various states.

Wilson has found that the brightest students work around what he calls their “unnecessary handicap.” In his study of 2007 Calculus III students, for example, the correlation between being “division clueless” and scoring poorly on the final exam wasn’t as strong as he would have guessed; some of those students did just fine. But that seemed true only for the minority. When he followed up on the class two years later in 2009, one-third of the “division-clueless” students were on academic probation.

Wilson doesn’t like the long-term implications of a new generation of engineers and scientists who can’t divide or don’t know their multiplication tables. He compares it to having car mechanics who only know how to fix automatic transmissions. “You might use a calculator if you’re an engineer, but you need to know what it does. If you need mathematics in your career, then it is probably a good idea to really understand it,” he says. “You don’t understand it if you can’t do it.”

Back to Basics for the “division clueless”
By Lisa Watts
December 6, 2010
Johns Hopkins Magazine

ChemProf said...

A couple years ago, I banned graphing calculators (which are highly programmable) from my class. It is amazing to see my students use basic scientific calculators. They enter unnecessary zeros (if a value is 1.000, they enter 1.000 not just 1). They don't really know their order of operations. They can't figure out how to enter numbers in scientific notation.

So why wasn't this a problem before? Well, some of it was a waste of time with the graphing calculator, but some of it was masked. With a graphing calculator, they can just enter the whole equation and the calculator evaluates it.

I've also seen the "division-clueless" effect. When I get a student who reaches for her calculator to figure out 10% of something, I can be pretty sure she won't complete a science major.

John Warner said...

Two observations from the classroom - firstly - the weaker the child in mathematics the more they "hug" a calculator and believe it as some sort of totemic god - it must be right the calculator told me so.

Calculators give you an answer - and do not bore you with the details of the process - the higher mathematics is based on exploring the details of how you do things and changing things. I used to have periodic lessons when no calculators were available and we would set ourselves questions of long multiplication and division and find ways to solve them. I used to collect methods as there are several ways to tackle these problems and some are very helpful in developing algebra skills - for learning about multiplying out brackets or dividing polynomials etc.

Anne Dwyer said...

I teach in major university in the math department and was surprised to find this year's report from Dr. Wilson in my inbox. He still does this exercise every semester and correlates the results to final grades. Then he sends his findings out to other people. I'm not sure if he sends it to other math departments or just to certain colleagues who pass it on.

This semester, I taught the same class (Intermediate Algebra) at a community college and at the university. The main difference was that the university did not allow calculators and the community college emphasized technology. I don't have any great conclusions. The really top students do well in either environment. The bottom students drop the class in either environment. I suspect, but don't have any evidence, that non calculator use benefits the students in the middle the most in their next classes.

SteveH said...

"Curriculum directors listen to education schools, workshops, and consultants. Not to mathematicians."

I've seen this in action. Our schools used MathLand and then switched to Everyday Math, ignoring a Citizen's Curriculum Committee in the process. Unfortunately, basic algorithms like long division seem to be the battleground. We end up arguing pedagogy and not curriculum and mastery of other important skills like fractions and percents. The reasons they use to avoid long division fall apart when you get to more advanced topics in math.

There is nothing wrong with calculators; just how they are used. They are used as avoidance tools when they should be used to make learning more interesting and rigorous. Rather than collecting data on what types of ice cream people like in the student's family (My son had that one and he had to draw pictures of cones in the form of a bar chart.) that had too little data to mean anything, they could have had the students use a calculator to work on a large data set already provided by the teacher. They wouldn't waste time collecting data and drawing pictures. The argument is really about low versus high expectations, not rote algorithms versus understanding.

This really hit home for me when my son was in fifth grade. I was trying to get the head of curriculum to include Singapore Math on the list of math curricula they were evaluating. I loaned her my books and gave my best, low pressure pitch. She knew I was an engineer and that I once taught college math and computer science.

The advantage of Singapore Math is that it's harder to reject out of hand than something like Saxon Math. In the end, however, she told me that although Singapore Math looked good, Everyday Math was better for "their mix of kids".

I was fighting ignorance. She didn't know math well. She undervalued mastery of the basic algorithms. She equated mastery with rote, no-understanding calculation. This fits nicely with their Ed school belief system. They see a student struggle to apply the wrong algorithm to a problem and educators come to the completely wrong conclusion; that somehow understanding has to come from a source other than mastery of an algorithm. They joyfully come to the conclusion that understanding and problem solving comes from top-down discovery and thematic learning.

This might be a solution if they could only get it to work. Unfortunately, they have deprecated mastery so much that schools leave it to chance. They talk of balance, but that doesn't mean that they will ensure that it happens.

I've said before that it boils down to a turf issue. One noted K-12 educator (I've lost the link.) once commented snidely that college professors need to keep in mind who is in charge.

Glen said...

She undervalued mastery of the basic algorithms. She equated mastery with rote, no-understanding calculation.

So true. I think part of the problem is that these people don't know many masters of calculation.

When I was a kid, I was always amazed by the old geezer engineers (WWII to Apollo Program era) and their ability to run a quick mathematical analysis of something in their heads as they discussed it. They could estimate anything -- trig functions, logs, powers, roots, even physical, chemical, financial, demographic data -- to one or two digits of precision plus a power of ten, quickly combine them, and reach a tentative conclusion.

Some of us who knew those guys have tried to emulate them, but most engineers of my era aren't good at it. They'll adjourn a meeting to go "run the numbers" on their computers, while the old guys would have been able to find the flaw in an idea during the meeting with a few seconds of hard THINKING.

To be fair, my generation is better with computers than the old guys were, making it possible for us to gain some (preliminary) understanding of such things as complex adaptive systems. The best engineers of today can do what the old guys did AND build computer models.

Now THAT'S conceptual understanding, but most educrats don't spend time with people like this and don't have any understanding of what they do or how they learned to do it. I'm afraid their approach will be more successful at producing future educrats than future engineers.

Michael Weiss said...

Curriculum directors listen to education schools, workshops, and consultants. Not to mathematicians.

What about mathematicians who work with, or in, education schools?

We do exist, you know.

Catherine Johnson said...

In my district, curriculum directors have been primarily English teachers. I think the curriculum director who chose Trailblazers was a social studies teacher.

My school doesn't seek advice from anyone at the college level & actively rejects advice from disciplinary specialists.

True story: a few years ago, Ed said to one of the building principals, "I have been a disciplinary specialist for 25 years."

She said, "Have you ever thought maybe that's your problem?"

Catherine Johnson said...

Hi Anne!

Interesting.

I think there's a passage in Wilson's article where he says there's a group of very talented students who don't seem to be affected by calculator use, which I think supports your impression --- (but tell me if not - )

PhysicistDave said...

Glen wrote:
>To be fair, my generation is better with computers than the old guys were, making it possible for us to gain some (preliminary) understanding of such things as complex adaptive systems. The best engineers of today can do what the old guys did AND build computer models.

Having worked for a number of years in engineering, I'd amend that to say, "Any competent engineer of today can do what the old guys did AND build computer models."

Obviously, you have to be able to use computer models. But, an ability to use computer models that is mot coupled with an intuitive understanding is, I think, a clear sign of a technically incompetent person.

And, I'd apply this beyond those who aspire to technical professions. The best understanding we have today of both the natural world and of the nature of human beings is based on science, and understanding science requires one to grasp mathematics.

No doubt many people manage to live self-satisfied lives in blissful ignorance of our best knowledge of reality. I'm also sure that many pet poodles are very satisfied with their lives.

A human being who chooses to live in ignorance of modern science is a person who chooses to allow his ideas of reality to be based on ignorance and to be molded by other, often dishonest and manipulative, human beings.

Dave Miller in Sacramento

ChemProf said...

I'll disagree with you, Dave. Part of this is engineering education has changed. My grandfather was an engineer in the "computer is a job description" era. When he designed the spillways at Hoover Dam, it took him six months to finish the calculations. Because the calcs took so long, engineers in that era knew a lot of short-cut estimations. Now that the calculations take a couple of minutes, those estimations aren't taught. Yes, a good engineer develops an intuitive number sense over time, but that isn't the same thing.

PhysicistDave said...

ChemProf wrote:
> I'll disagree with you, Dave. Part of this is engineering education has changed. My grandfather was an engineer in the "computer is a job description" era. When he designed the spillways at Hoover Dam, it took him six months to finish the calculations. Because the calcs took so long, engineers in that era knew a lot of short-cut estimations. Now that the calculations take a couple of minutes, those estimations aren't taught.

Yeah, but that is not a good thing, for a variety of reasons. For example, having an intuitive sense for the calculations gives one a good feel for what calculations you should actually bother to have the computer do, what sort of design options are likely to prove feasible and are worth pursuing in more detail, what results are likely to prove robust under various real-world perturbations, etc.

I’ll give two examples of how bad the situation has gotten from my own work experience – both are from some years ago (the change seems to have come with guys educated after the early seventies when hand-held calculators became widespread).

In one case, we had a problem where we had to multiply 1.9 times 2.1. One of us older guys (i.e., we got our BS before 1980) causally mentioned that the answer would be about 4. A young engineer, just out of UCLA with an “A” average, pulled out his calculator and found that the answer is 3.99. He turned to us older guys and asked with amazement how we knew that it was about 4. (Yeah, we patiently explained it.)

Another case: a colleague had patiently taken data on a circuit problem and gotten results of the form 1.01 Volt vs. 0.99 Amps, 2.1 V vs. 1.11 A, etc. (I don’t remember the exact values, but they were in fact very, very close to integers). He had shown this data around to several engineers, and no one could figure out what it meant. I immediately saw that it meant his circuit had a resistance of about 1 Ohm – i.e., he had a short circuit.

A bright high-school student (okay, a bright middle-school student) should have found both of these problems to be as easy as pie. Yet, guys with BS’s in engineering, out in the working world, could not deal with them.

You’re probably right that this is a result of a change in engineering education. It is not a good change.

Dave

ChemProf said...

Sorry if I wasn't clear -- I'd absolutely agree that it wasn't a good change, but was noting there was a reason that even well educated young engineers don't have the basic sense that the older guys did.

I was fortunate to be educated on the edge of the calculator era (got my first one in high school chem because I was finishing exams first despite being the only student without one). My students are shocked that I can add multi digit numbers in my head, and they are often slower than I am with their calculators. I hate to say that none of your stories surprise me!

PhysicistDave said...

ChemProf wrote:
> Sorry if I wasn't clear -- I'd absolutely agree that it wasn't a good change, but was noting there was a reason that even well educated young engineers don't have the basic sense that the older guys did.

Yeah, I understand your point now.

Incidentally, I did not get a calculator until my freshman year of college (sounds like I’m probably a couple years older than you), so that I was already in the habit of, at least, doing approximations in my head.

You’ll probably agree with me that the engineering profs did not intend to produce this development: it was just easier to let the students go ahead and use the calculators. And, of course, in the real world, engineers do use calculators and computers all the time.

Nonetheless, it was a failure of pedagogy: the engineering professors should have limited calculator use, tested on knowledge of all the old intuitive rules of thumb, etc. Their students would have become much better engineers (who would, of course, still have used calculators and computers in the real world to get the last decimal place).

Even those of us who are good at mental math and estimation and who strive to understand what is going on intuitively can make serious mistakes on occasion: I’ve made some whoppers myself, most of which, fortunately, were caught by colleagues before they actually wrecked our projects.

It’s hard enough to get difficult technical projects done right even with bright, diligent, well-trained people – there are just too many ways to foul up. When the educational system fails to give the best training possible, we turn out technical people who, in effect, have one arm tied behind their back.

Of course, as Glen suggested, you will always have a few guys who transcend the system, who are so obsessively brilliant in their field that they can do a great job regardless of the shortcomings of the educational system. But everyone else is really short-changed.

Dave

P.S. Hope that you (and everyone here) had a Merry Christmas and that you have a Happy New Year!

Glen said...

Me: The best engineers of today can do what the old guys did AND build computer models.

Dave: Having worked for a number of years in engineering, I'd amend that to say, "Any competent engineer of today can do what the old guys did AND build computer models."

All three of us are clearly on the same page in every respect here, except that I think I'll keep my original statement instead of your restatement, Dave. This is why: I wasn't talking about engineers who can't "feel" that a smidge above 2 times a smidge below 2 is about 4. This is an excellent example of the perils of a calculator curriculum, and I WOULD accept your amendment regarding such cases.

But I was talking about old engineers who routinely estimated trig, log, power, root, etc. functions, and so on [see my orig post above]. They were on a whole different plane. I think this was routine for the Old Ones, but is something "only the best" do today.

I'm not ready to accuse an engineer of incompetence if he can't quickly estimate log(35)/sin(20 deg) to one significant digit plus a power of ten. I think he should be able to, and most of the old guys could, but I won't call him incompetent if he can't. I'll just leave it at saying that only the best today can do what was routine fifty years ago with respect to the level of math sense I'm talking about.

But I do feel that an engineer (or scientist) ought to be able to do these things if he aspires to mastery of his field, and, well, yeah, okay, it might not take too much persuasion to get me to confess that I would at least wonder about the competence of one who couldn't estimate any numerical calculation beyond basic arithmetic.

PhysicistDave said...

Glen wrote:
>All three of us are clearly on the same page…

Yeah, my rephrasing was just emphasizing your point.

Glen also wrote:
>I'm not ready to accuse an engineer of incompetence if he can't quickly estimate log(35)/sin(20 deg)…

Well, let’s see, by mental math, I get a value of (taking “log” to refer to log to the base ten) 4.33; if you mean natural log, I get 10.1. Checking on the calculator, I get 4.51 and 10.4.

So, yeah, I guess I *do* think a technically competent person should be able to do this in his head to, say, ten percent accuracy (admittedly, you threw out an easy one: if you’d asked for log(87)/sin(35), my accuracy would have been somewhat lower).

I’ll admit that I do know competent engineers whose estimates would be further off here than mine, but it really is not that hard to get that the answer is between 3 and 5. My really quick and dirty estimate actually would have been 4.5, which is very close to the correct answer, largely by accident: obviously 35 is close to 31.62, and log10 (31.62) is 1.5, and 20 deg is about a third of a radian, so sine(20 deg) is around a third – 1.5 over a third is 4.5.

Anyone who cannot follow that last sentence, in my judgment really is not technically competent. (It is amusing that my really sloppy estimate happens to be slightly more accurate than my more careful mental-math estimate.)

Glen also wrote:

I do agree that this is the more important point – a good, experienced engineer does not really have to do mental math to get within a factor of two or so: he or she just “sees” what is happening. Misuse of calculators and computers can be a real killer on acquiring that skill.

Let me reiterate of course that none of us are arguing that technical people should never use calculators or computers – they are great for getting an extra decimal point and for checking our intuition and mental math (even someone good at mental math can slip a decimal point or forget a factor of two: humans make errors). The problem is letting the machines serve as an excuse for not having a deep understanding.

As Liping Ma says, a “profound understanding of fundamental mathematics” is the goal. And, I’d amend that to “a profound understanding of anything in which you claim expertise.” I don’t have a profound understanding of chemistry, for example, as some of my exchanges with ChemProf have, alas, shown, even though I got good grades in HS and college chem. But, I hope I do have a profound understanding of the areas of physics and engineering in which I have worked.

Dave

SteveH said...

'She said, "Have you ever thought maybe that's your problem?" '

I've heard this sort of thing before. It works great for them. People who are very good in a subject make the worst teachers because they probably didn't have to struggle with the material; they can't understand or empathize with the average student. A better approach is to just tell educators that they are teaching the wrong things using the wrong techniques. In math, tell them that that they are not being supportive. Tell them that they are making it impossible for many kids to ever have a STEM career. That probably won't work either. You will then be accused of being part of the problem and not part of the solution.

SteveH said...

I was a junior in college when I bought my first calculator. I saw everything change - for the better. Calculations became more complex and we could do more than just simple theories that were limited by the sliderule. Exams had more symbolic manipulation rather than numeric calculations.

Now that I'm working on a contract through my old department at UofMichigan, I'm getting some feedback from my old adviser. He has seen over the last forty years a slow watering-down of the math requirements for engineering. It's not so much just a matter of numeric estimation, but of everything. Students who did well in AP calculus are struggling. It's a big problem for the departments. They are losing students after the first year. They have meetings to decide how to support these students. The Engineering College is looking at more hands-on approaches to engage or motivate the students. Once again, educators see a problem and come to the wrong conclusion. It's not an issue of motivation. It's an issue of how the students are struggling to keep their mathematical heads above water.

If you have to integrate a load curve to get a shear curve, and then integrate that to get a bending moment curve, you can't be struggling with the basics of integration. You have other things to learn. You might have more students who know how to use a computer analysis program, but fewer who could write one.

ChemProf said...

The estimation ability is just the canary in the coal mine. It is like knowing your multiplication facts. My students can't estimate logs because they don't know what a logarithm IS, just how to punch the numbers on the calculator. Because of pH, I spend a chunk of time talking about what logs are for, and it is news to almost all of my students.

When I learned about sines and cosines, in the early calculator days, we spent a good week drawing the curves, and calculating the point every 10 degrees or so for different equations. Now that is a one day activity with a graphing calculator, and seems more efficient, but it doesn't leave students with a feel for what values they should get.

Then, they hit calculus with all of these built up deficits, and don't quite master calculus. Since they haven't mastered basic integration of complex functions, real engineering integration is beyond them. It all builds up.

More and more, as the math is watered down, those of us in science and engineering take up the slack, but that isn't the optimal solution. I am starting to teach thermodynamics in a few months, and plan to start with a two day version of "seat-of-the-pants" multivariable, since most of my students won't have it (despite what the course descriptions say).

Glen said...

Dave, sorry, I was ambiguous with "log()". When I'm doing calculus, I think in terms of natural log, but when doing engineering estimates, I usually assume common logs, because it's so easy to convert in and out of scientific notation. Machines can work in any base, but common logs used to be primarily a tool for humans to think with.

Which brings me to ChemProf's:The estimation ability is just the canary in the coal mine. It is like knowing your multiplication facts. My students can't estimate logs because they don't know what a logarithm IS...

Right, since one of the principal uses of logs has always been estimation of powers, roots, etc., if you don't do such estimation, you're much less likely to have a feeling for logs. Then when you need them for some other purpose (pH, carbon dating, etc.), you are more likely to have to simply memorize the formulae, because you can't see what the formulae mean. And I think you have a harder time understanding the phenomena being described by the formulae. They become black boxes, like your calculator.

PhysicistDave said...

SteveH wrote:
>I was a junior in college when I bought my first calculator. I saw everything change - for the better. Calculations became more complex and we could do more than just simple theories that were limited by the sliderule. Exams had more symbolic manipulation rather than numeric calculations.

Sure – but by that point you were already ~20 years old and had learned to *think* numerically: the caluclators were saving you scutwork (good), not preventing you from thinking (not so good). Incidentally, your timeline implies you’re of comparable age to me and ChemProf.

SteveH also wrote:
>He has seen over the last forty years a slow watering-down of the math requirements for engineering. It's not so much just a matter of numeric estimation, but of everything. Students who did well in AP calculus are struggling.

No disagreement there.

My favorite example is the failure to teach the axiomatic method in most HS geoemtry courses nowadays: see the comment a decade ago from then-chair of the Caltech math department, Barry Simon, at http://www.math.caltech.edu/people/oped.html . The problem is not of course that most scientists or engineers routinely use the axiomatic method in our daily work, but rather that someone who has not learned it is cut off from really learning most higher math from the mathematicians (not to mention the cultural and historical loss).

Incidentally, we should make clear that just as there are some whiz kids who transcend the system, there are of course some teachers who do fight back against the system. I have mentioned the HS student I know who is taking AP calc this year: her course is *very* challenging – I would have found it demanding as a HS student myself. The best students and the best teachers transcend the system. But they are of course rare.

Dave

PhysicistDave said...

Glen wrote:
>Dave, sorry, I was ambiguous with "log()". When I'm doing calculus, I think in terms of natural log, but when doing engineering estimates, I usually assume common logs, because it's so easy to convert in and out of scientific notation.

Yeah, I suppose that is true of pretty much everyone.

Glen also wrote:
>Right, since one of the principal uses of logs has always been estimation of powers, roots, etc., if you don't do such estimation, you're much less likely to have a feeling for logs. Then when you need them for some other purpose (pH, carbon dating, etc.), you are more likely to have to simply memorize the formulae, because you can't see what the formulae mean. And I think you have a harder time understanding the phenomena being described by the formulae. They become black boxes, like your calculator.

Yeah, that is really the point – all of us have the “logs grow really slowly,” “exponentials really blow up fast,” etc. sort of intuition. Without all of that intuition, you do not know what you are doing.

Of course, teaching that sort of intuition is hard. But a teacher unwilling to at least try should just hand over the textbooks to the kids and leave the room.

Dave

SteveH said...

"Sure – but by that point you were already ~20 years old and had learned to *think* numerically"

Physicist Dave: perhaps you missed this comment I made in an earlier post:

There is nothing wrong with calculators; just how they are used. They are used as avoidance tools when they should be used to make learning more interesting and rigorous.

K-12 schools could use calculators properly if they wanted. Just because they don't doesn't mean that there is a problem with the calculator. There is only a problem with educators. It also means that the argument is not about understanding or algorithms. It's about low expectations. K-8 educators try to focus the conversation on understanding and critical thinking, but the real issue is that they are redefining math and lowering expectations.

Years ago I called this brain research misdirection. Educators like to define the problem in terms of fancy concepts of science and pedagogy when in reality, it's all about low expectations. Even if I were to accept their ideas of pedagogy and learning styles, the details show that their expectations are very low.

PhysicistDave said...

SteveH wrote:
>There is nothing wrong with calculators; just how they are used. They are used as avoidance tools when they should be used to make learning more interesting and rigorous.
>K-12 schools could use calculators properly if they wanted. Just because they don't doesn't mean that there is a problem with the calculator. There is only a problem with educators.

Indeed. On the other hand, “properly” using calculators in some situations – i.e., when kids are still learning the fundamental algorithms – is so hard that the best solution is probably just to ban them. I really do not think a second-grader should be using a calculator in her schoolwork at all.

SteveH also wrote:
>It also means that the argument is not about understanding or algorithms. It's about low expectations. K-8 educators try to focus the conversation on understanding and critical thinking, but the real issue is that they are redefining math and lowering expectations.

I actually think that “low expectations” is only part of it. I think that most grade-school teachers and even a fair number of high-school math and science teachers (I have in mind my own HS physics teacher, who should not have been allowed to pass a HS physics course himself!) have so little real understanding of math, or of the understanding of math that is required in technical fields, that it is nearly impossible for them to avoid “low expectations.” (I assume this does not apply to college instructors, almost all of whom – I hope! – are numerate.)

Again, I appeal to Liping Ma’s research in her Knowing and Teaching Elementary Mathematics: her findings match what I have seen myself. And the willingness of HS math teachers to abandon the axiomatic method also does not speak well for their understanding of real math.

Of course, as you know, my main theme is that the system is broken in so many ways that trying to repair it is worse than hopeless. But, I do not see that as a cause for pessimism: when your car is such a clunker that it is clear it can never be fixed, you do not sit down and weep; you simply acknowledge that the time has come to treat yourself to a new car. Similarly, seeing how wrecked our official educational system is should not be a cause for despair but merely a welcome reason to abandon the system altogether.

Dave

SteveH said...

'...that it is nearly impossible for them to avoid “low expectations.” '

Our state requires content certification starting in 7th grade for all subjects. I saw a distinct change when my son got to that point. It wasn't great, but he started to have real math with real textbooks. Unfortunately, it's too late for many kids.

I'm not sure what "abandon the system altogether" means or how that would work.

Although there are many issues that could be solved, I think that the worst of them lie in K-8. It may seem backwards, but a number of parents (us included) send their kids to a private school in K-8 and then back to the public school in high school. We ended up bringing our son back in sixth grade because the private school had their own kind of fuzzy educational issues and we were able to manage in the public school ... and save \$15,000+ a year.

Any alternatives to regular public schools face cost issues, ed school issues, and home court advantages. The biggest one is that no matter where you go, you will run into ed school philosophy. With charter schools, you might get lucky, but most all of them in our area have very odd charters and go towards hands-on, fuzzy ideas of education. Nobody is jumping up to create Core Knowledge charter schools. (They would be rejected by the state anyways.) Also, the big problem that we ran into is that my son didn't want to spend an hour and a half each day commuting to school. Compare that to the 5 minute walk to our public school.

Catherine Johnson said...

'She said, "Have you ever thought maybe that's your problem?" '

I've heard this sort of thing before. It works great for them. People who are very good in a subject make the worst teachers because they probably didn't have to struggle with the material; they can't understand or empathize with the average student.

It was worse than that!

She opposes teaching the liberal arts disciplines as disciplines.

All subjects should be taught as One.

She believes that it's wrong to teach content "in isolation."

She's a wholeist.

Catherine Johnson said...

Remember histogeomegraph?

Catherine Johnson said...

And whole math?

PhysicistDave said...

SteveH wrote to me:
> I'm not sure what "abandon the system altogether" means or how that would work.

You’re not? Really?

I mean shut it down. Completely.

We do not have a “public” (i.e., government-owned, -operated, and –financed) clothing system or food system, etc. Parents provide food, clothing, etc. for their own kids themselves. Obviously, education can work the same way, and, indeed, did work the same way throughout most of human history.

I realize most parents say they just could not provide education themselves for their own kids. But, suppose that some horrible plague required us to shut down the schools for a few years to retard the spread of the plague. Somehow, parents would indeed manage to provide their own kids with education, just as they now provide them with food and clothing.

So, they could do it now, if they really wanted to. They could homeschool, form small educational cooperatives, or whatever. Just get their kids out of the schools.

Most parents do not do that because they are really not all that dissatisfied with the public schools. Don’t get me wrong: I think everyone should be not simply dissatisfied but deeply horrified with the public schools; I think the public schools should be viewed with the same horror with which we would view open pedophiles.

But most parents do not have that perspective at all. They “can’t” take their kids out of the schools because they value other things more highly in terms of the use of their own time, energy, and money.

Not much I can do about their personal choices. But my personal advice is that they should quite literally abandon the system.

I meant literally what I said: I’m a little surprised you weren’t sure what I meant.

Dave