How to decide if anybody should listen to your ideas on how and whether to re-open schools, or maybe you should just hush.

Peter Greene has provided a nice flow chart to let you decide whether you should open your mouth with your ideas on how and whether to re-open the public schools, or whether you should just be quiet and listen.

So, should you just hush, or do you have something valuable to contribute to this subject?

My wife and I each taught for 30 years or so, and so we would be in the ‘speak right up’ category, but I don’t really know how the USA can get public education to work next year, especially since the danger is not going away, but apparently once more growing at an exponential clip.

Nobody should be listening to billionaires or their bought-and-paid-for policy wonks who once spent a whole two years in a classroom.

A few quotes from Greene’s column. (He is a much better writer than me, and much more original as well.)

==================================

To everyone who was never a classroom teacher but who has some ideas about how school should be reopened in the fall:

Hush.

Just hush.

There are some special categories of life experiences. Divorce. Parenthood. Deafness. Living as a Black person in the US. Classroom teacher. They are very different experiences, but they all have on thing in common.

You can read about these things. But if you haven’t lived it, you don’t know. You can study up, read up, talk to people. And in some rare cases that brings you close enough to knowing that your insights might actually be useful.

But mostly, you are a Dunning-Krueger case study just waiting to be written up.

The last thirty-seven-ish years of education have been marked by one major feature– a whole lot of people who just don’t know, throwing their weight around and trying to set the conditions under which the people who actually do the work will have to try to actually do the work. Policy wonks, privateers, Teach for America pass-throughs, guys who wanted to run for President, folks walking by on the street who happen to be filthy rich, amateurs who believe their ignorance is a qualification– everyone has stuck their oar in to try to reshape US education. And in ordinary times, as much as I argue against these folks, I would not wave my magic wand to silence them, because 1) educators are just as susceptible as anyone to becoming too insular and entrenched and convinced of their own eternal rightness and 2) it is a teacher’s job to serve all those amateurs, so it behooves the education world to listen, even if what they hear is 98% bosh.

But that’s in ordinary times, and these are not ordinary times.

There’s a whole lot of discussion about the issues involved in starting up school this fall. The discussion is made difficult by the fact that all options stink. It is further complicated by the loud voices of people who literally do not know what they are talking about.

Will these ‘lost’ months of school really matter?

David Berliner explains that the academic topics untaught during these months of coronavirus shutdowns of schools aren’t really all that much to worry about — as long as kids have been engaged in useful or imaginative projects of their own choosing. This first appeared on Diane Ravitch’s blog. I found it at Larry Cuban’s blog.

Worried About Those “Big” Losses on School Tests Because Of Extended Stays At Home? They May Not Even Happen,
And If They Do, They May Not Matter Much At All!

David C. Berliner
Regents Professor Emeritus
Mary Lou Fulton Teachers College
Arizona State University
Tempe, AZ.

Although my mother passed away many years ago, I need now to make a public confession about a crime she committed year in and year out. When I was young, she prevented me from obtaining one year of public schooling. Surely that must be a crime!

Let me explain. Every year my mother took me out of school for three full weeks following the Memorial Day weekend. Thus, every single year, from K through 9th grade, I was absent from school for 3 weeks. Over time I lost about 30 weeks of schooling. With tonsil removal, recurring Mastoiditis, broken bones, and more than the average ordinary childhood illnesses, I missed a good deal of elementary schooling.
How did missing that much schooling hurt me? Not at all!

First, I must explain why my mother would break the law. In part it was to get me out of New York City as the polio epidemic hit U.S. cities from June through the summer months. For each of those summers, my family rented one room for the whole family in a rooming house filled with working class families at a beach called Rockaway. It was outside the urban area, but actually still within NYC limits.

I spent the time swimming every day, playing ball and pinochle with friends, and reading. And then, I read some more. Believe it or not, for kids like me, leaving school probably enhanced my growth! I was loved, I had great adventures, I conversed with adults in the rooming house, I saw many movies, I read classic comics, and even some “real” literature. I read series after series written for young people: Don Sturdy, Tom Swift, the Hardy Boys, as well as books by Robert Louis Stevenson and Alexander Dumas.

So now, with so many children out of school, and based on all the time I supposedly lost, I will make a prediction: every child who likes to read, every child with an interest in building computers or in building model bridges, planes, skyscrapers, autos, or anything else complex, or who plays a lot of “Fortnite,” or “Minecraft,” or plays non-computer but highly complex games such as “Magic,” or “Ticket to Ride,” or “Codenames” will not lose anything measurable by staying home. If children are cared for emotionally, have interesting stuff to play with, and read stories that engage them, I predict no deficiencies in school learning will be detectable six to nine months down the road.
It is the kids, rich or poor, without the magic ingredients of love and safety in their family, books to engage them, and interesting mind-engaging games to play, who may lose a few points on the tests we use to measure school learning. There are many of those kinds of children in the nation, and it is sad to contemplate that.

But then, what if they do lose a few points on the achievement tests currently in use in our nation and in each of our states? None of those tests predict with enough confidence much about the future life those kids will live. That is because it is not just the grades that kids get in school, nor their scores on tests of school knowledge, that predict success in college and in life. Soft skills, which develop as well during their hiatus from school as they do when they are in school, are excellent predictors of a child’s future success in life.

Really? Deke and Haimson (2006), working for Mathmatica, the highly respected social science research organization, studied the relationship between academic competence and some “soft” skills on some of the important outcomes in life after high school. They used high school math test scores as a proxy for academic competency, since math scores typically correlate well with most other academic indices. The soft skills they examined were a composite score from high school data that described each students’ work habits, measurement of sports related competence, a pro-social measure, a measure of leadership, and a measure of locus of control.

The researchers’ question, just as is every teacher’s and school counselor’s question, was this: If I worked on improving one of these academic or soft skills, which would give that student the biggest bang for the buck as they move on with their lives?

Let me quote their results (emphasis by me [-not me! GFB])

Increasing math test scores had the largest effect on earnings for a plurality of the students, but most students benefited more from improving one of the nonacademic competencies. For example, with respect to earnings eight years after high school, increasing math test scores would have been most effective for just 33 percent of students, but 67 percent would have benefited more from improving a nonacademic competency. Many students would have secured the largest earnings benefit from improvements in locus of control (taking personal responsibility) (30 percent) and sports-related competencies (20 percent). Similarly, for most students, improving one of the nonacademic competencies would have had a larger effect than better math scores on their chances of enrolling in and completing a postsecondary program.

​This was not new. Almost 50 years ago, Bowles and Gintis (1976), on the political left, pointed out that an individual’s noncognitive behaviors were perhaps more important than their cognitive skills in determining the kinds of outcomes the middle and upper middle classes expect from their children. Shortly after Bowles and Gintis’s treatise, Jencks and his colleagues (1979), closer to the political right, found little evidence that cognitive skills, such as those taught in school, played a big role in occupational success.

Employment usually depends on certificates or licenses—a high school degree, an Associate’s degree, a 4-year college degree or perhaps an advanced degree. Social class certainly affects those achievements. But Jenks and his colleagues also found that industriousness, leadership, and good study habits in high school were positively associated with higher occupational attainment and earnings, even after controlling for social class. It’s not all about grades, test scores, and social class background: Soft skills matter a lot!

Lleras (2008), 10 years after she studied a group of 10th grade students, found that those students with better social skills, work habits, and who also participated in extracurricular activities in high school had higher educational attainment and earnings, even after controlling for cognitive skills! Student work habits and conscientiousness were positively related to educational attainment and this in turn, results in higher earnings.

It is pretty simple: students who have better work habits have higher earnings in the labor market because they are able to complete more years of schooling and their bosses like them. In addition, Lleras’s study and others point to the persistent importance of motivation in predicting earnings, even after taking into account education. The Lleras study supports the conclusions reached by Jencks and his colleagues (1979), that noncognitive behaviors of secondary students were as important as cognitive skills in predicting later earnings.
So, what shall we make of all this? I think poor and wealthy parents, educated and uneducated parents, immigrant or native-born parents, all have the skills to help their children succeed in life. They just need to worry less about their child’s test scores and more about promoting reading and stimulating their children’s minds through interesting games – something more than killing monsters and bad guys. Parents who promote hobbies and building projects are doing the right thing. So are parents who have their kids tell them what they learned from watching a PBS nature special or from watching a video tour of a museum. Parents also do the right thing when they ask, after their child helps a neighbor, how the doing of kind acts makes their child feel. This is the “stuff” in early life that influences a child’s success later in life even more powerfully than do their test scores.

So, repeat after me all you test concerned parents: non-academic skills are more powerful than academic skills in life outcomes. This is not to gainsay for a minute the power of instruction in literacy and numeracy at our schools, nor the need for history and science courses. Intelligent citizenship and the world of work require subject matter knowledge. But I hasten to remind us all that success in many areas of life is not going to depend on a few points lost on state tests that predict so little. If a child’s stay at home during this pandemic is met with love and a chance to do something interesting, I have little concern about that child’s, or our nation’s, future.

Bowles, S., & Gintis, H. (1976). Schooling in Capitalist America. New York: Basic Books.

Deke, J. & Haimson, J. (2006, September). Expanding beyond academics: Who benefits and how? Princeton NJ: Issue briefs #2, Mathematica Policy Research, Inc. Retrieved May 20, 2009 from:http://www.eric.ed.gov:80/ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/28/09/9f.pdfMatematicapolicy research Inc.

Lleras, C. (2008). Do skills and behaviors in high school matter? The contribution of noncognitive factors in explaining differences in educational attainment and earnings. Social Science Research, 37, 888–902.

Jencks, C., Bartlett, S., Corcoran, M., Crouse, J., Eaglesfield, D., Jackson, G., McCelland, K., Mueser, P., Olneck, M., Schwartz, J., Ward, S., and Williams, J. (1979). Who Gets Ahead?: The Determinants of Economic Success in America. New York: Basic Books.

 

The Pandemic Is Far From Over

While the rate of increase per day in the number of deaths is generally down, the COVID-19 pandemic is far from over. In general, more people are still dying each day in the US from this disease than the day before, as you can see from this data, which is taken from the CDC. The very tall bar on day 27 is when New York City finally added thousands of poor souls who had in fact died from this virus. (Day 27 means April 9, and Day 41 means April 30, which is today.)

Opening up the economy and encouraging everybody to go back to work, play, and school will mean a rebirth of exponential growth in deaths and in diagnosed cases after about 2 weeks, since this disease takes about that long to be noticed in those who have been exposed. And once everybody is back on the streets and in the stores and schools, the disease WILL spread exponentially. Opening wide right now, when we still can’t test or follow those who may be infected, would be a huge mistake.

us covid deaths per day

Only somebody as clueless as our current Grifter-In-Chief and his brainless acolytes could be recommending something so irresponsible, against the advice of every medical expert. Maybe they think that only the poor, the black, and the brown will get this disease. Wrong.

The shutdown, while painful, appears to have saved a LOT of lives so far

If you recall, the growth of the new corona virus disease in the US (and many other countries) at first looked to be exponential, meaning that the number of cases (and deaths) were rising at an alarming, fixed percent each and every single day.

Even if you slept through your high school or middle school math lessons on exponential growth, the story of the Shah and the chessboard filled with rice may have told you that the equation 2^x gets very, very hairy after a while. Pyramid schemes eventually run out of suckers people. Or perhaps you have seen a relatively modest credit-card bill get way out of hand as the bank applies 8 percent interest PER MONTH, which ends up multiplying your debt by a factor of 6 after just 2 years!

(If the total number of deaths were still increasing by 25 percent per day, as they were during the middle of March, and if that trend somehow continued without slowing down, then every single person residing inside America’s borders would be dead before the end of May. Not kidding! But it’s also not happening.)

However, judging by numbers released by the CDC and reported by my former colleague Ron Jenkins, I am quite confident that THE NUMBER OF CASES AND DEATHS FROM COVID-19 ARE NO LONGER following a fixed exponential curve. Or at least, the daily rate of increase has been going down. Which is good. But it’s still not zero.

Let me show you the data and fitted curves in a number of graphs, which often make complex things easier to visualize and understand.

My first graph is the total reported number of deaths so far in the US, compared to a best-fit exponential graph:

Deaths in US are not growing exponentially

During the first part of this pandemic, during the first 40 or so days, the data actually fit an exponential graph pretty well – that is, the red dotted line (the exponential curve of best fit) fit the actual cumulative number of deaths (in blue). And that’s not good. However, since about day 50 (last week) the data is WAY UNDER the red dots. To give you an idea of how much of a victory that is: find day 70, which is May 9, and follow the vertical line up until it meets the red dotted line. I’ll wait.

Did you find it? If this pandemic were still following exponential growth, now and into the future, at the same rate, we would have roughly a MILLION PEOPLE DEAD BY JUNE 9 in just the US, just from this disease, and 2 million the week after that, and 4 million the next week, then 8 million, then 16 million, and so on.

THAT AIN’T HAPPENIN’! YAY! HUZZAH!

As you can see — the blue and red graphs have diverged. Ignore the relatively high correlation value of 0.935 – it just ain’t so.

But what IS the curve of best fit? I don’t know, so I’ll let you look for yourself.

Is it linear?

Deaths in US are not growing in a linear fashion

This particular line of best doesn’t fit the data very well; however, if we start at day 36 or thereabouts, we could get a line that fits the data from there on pretty well, like so:

maybe this purple line

 

The purple line fits the blue dots quite well after about day 37 (about April 6), and the statistics algorithms quite agree. However, it still calls for over 80,000 Americans dead by May 8. I do not want the slope of that line to be positive! I want it to turn to the right and remain horizontal – meaning NOBODY ELSE DIES ANY MORE FROM THIS DISEASE.

Perhaps it’s not linear? Perhaps it’s one of those other types of equations you might remember from some algebra class, like a parabola, a cubic, or a quartic? Let’s take a look:

Deaths might be growing at a 2nd degree polynomial rate - still not good

This is a parabolic function, or a quadratic. The red dots do fit the data pretty well. Unfortunately, we want the blue dots NOT to fit that graph, because that would, once again, mean about a hundred thousand people dead by May 8. That’s better than a million, but I want the deaths to stop increasing at all. Like this piecewise function (which some of you studied). Note that the purple line cannot go back downwards, because generally speaking, dead people cannot be brought back to life.

maybe this purple line - nah, prefer horizontal

Well, does the data fit a cubic?

deaths fit a cubic very well

Unfortunately, this also fits pretty well. If it continues, we would still have about a hundred thousand dead by May 8, and the number would increase without limit (which, fortunately, is impossible).

How about a quartic (fourth-degree polynomial)? Let’s see:

4th degree polynomial is impossible - people do NOT come back to life

I admit that the actual data, in blue, fit the red calculated quartic red curve quite well, in fact, the best so far, and the number of deaths by Day 70 is the lowest so far. But it’s impossible: for the curve to go downwards like that would mean that you had ten thousand people who died, and who later came back to life. Nah, not happening.

What about logarithmic growth? That would actually be sweet – it’s a situation where a number rises quickly at first, but over time rises more and more slowly. Like this, in red:

logarithmic growth

I wish this described the real situation, but clearly, it does not.

One last option – a ‘power law’ where there is some fixed power of the date (in this case, the computer calculated it to be the date raised to the 5.377 power) which explains all of the deaths, like so:

no sign of a power law

I don’t think this fits the data very well, either. Fortunately. It’s too low from about day 38 to day 29, and is much too high from day 50 onwards. Otherwise we would be looking at about 230,000 dead by day 70 (May 8).

But saying that the entire number of deaths in the US is no longer following a single exponential curve doesn’t quite do the subject justice. Exponential growth (or decay) simply means that in any given time period, the quantity you are measuring is increasing (or decreasing) by a fixed percentage (or fraction). That’s all. And, as you can see, for the past week, the daily percentage of increase in the total number of deaths has been in the range of three to seven percent. However, during the first part of March, the rate of increase in deaths was enormous: 20 to 40 percent PER DAY. And the daily percent of increase in the number of cases was at times over A HUNDRED PERCENT!!! – which is off the chart below.

daily percentages of increases in covid 19 cases and deaths, USA, thru April 25

The situation is still not good! If we are stuck at a daily increase in the number of deaths as low as a 3%/day increase, then we are all dead within a year. Obviously, and fortunately, that’s probably not going to happen, but it’s a bit difficult to believe that the math works out that way.

But it does. Let me show you, using logs.

For simple round numbers, let’s say we have 50,000 poor souls who have died so far from this coronavirus in the USA right now, and that number of deaths is increasing at a rate of 3 percent per day. Let’s also say that the US has a population of about 330 million. The question is, when will we all be dead if that exponential growth keeps going on somehow? (Fortunately, it won’t.*) Here is the first equation, and then the steps I went through. Keep in mind that a growth of 3% per day means that you can multiply any day’s value by 1.03, or 103%, to get the next day’s value. Here goes:

in 10 months we are all dead

Sound unbelievable? To check that, let us take almost any calculator and try raising the expression 1.03 to the 300th power. I think you’ll get about 7098. Now take that and multiply it by the approximate number of people dead so far in the US, namely 50,000. You’ll get about 355,000,000 – well more than the total number of Americans.

So we still need to get that rate of increase in fatalities down, to basically zero. We are not there yet. With our current highly-incompetent national leadership, we might not.

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* what happens in cases like this is you get sort of an s-shaped curve, called the Logistic or logit curve, in which the total number levels off after a while. That’s shown below. Still not pleasant.

I have no idea how to model this sort of problem with a logistic curve; for one thing, one would need to know what the total ‘carrying capacity’ – or total number of dead — would be if current trends continue and we are unsuccessful at stopping this virus. The epidemiologists and statisticians who make models for this sort of thing know a lot more math, stats, biology, and so on than I do, but even they are working with a whole lot of unknowns, including the rate of infectiousness, what fraction of the people feel really sick, what fraction die, whether you get immunity if you are exposed, what is the effect of different viral loads, and much more. This virus has only been out for a few months…

logistic curve again

 

What’s the best approach – should we lock down harder, or let people start to go back to work? Some countries have had lockdowns, others have not. How will the future play out? I don’t know. I do know that before we can decide, we need to have fast, plentiful, and accurate tests, so we can quarantine just the people who are infected or are carriers, and let everybody else get back on with their lives. We are doing this lockdown simply because we have no other choice.

How do we fix the CV19 testing problem? By re-testing everybody who tested positive!

I guess I’ve re-discovered a form of Bayes’ Theorem  regarding the problem that is posed by the high numbers of false negatives and false positives when testing for the feared coronavirus.  What I found is that it doesn’t really even matter whether our tests are super-accurate or not. The solution is to assume that all those who test negative, really are negative, and then to give a second test to all those who tested positive the first time. Out of this group, a larger fraction will test positive. You can again forget about those who test negative. But re-test again, and if you like, test again. By the end of this process, where each time you are testing fewer people, then you will be over 99% certain that all those who test positive, really have been exposed.

Let me show you why.

Have no fear, what I’m gonna do is just spreadsheets. No fancy math, just percents. And it won’t really matter what the starting assumptions are! The results converge to almost perfect accuracy, if repeated!

To start my explanation, let’s start by assuming that 3% of a population (say of the US) has antibodies to CV19, which means that they have definitely been exposed. How they got exposed is not important for this discussion. Whether they felt anything from their exposure or not is not important in this discussion. Whether they got sick and died or recovered, is not going to be covered here. I will also assume that this test has a 7% false positive rate and a 10% false negative rate, and I’m going to assume that we give tests AT RANDOM to a hundred thousand people (not people who we already think are sick!) I’m also assuming that once you have the antibodies, you keep them for the duration.

This table represents that situation:

math of CV19 testing

If you do the simple arithmetic, using those assumptions, then of the 100,000 people we tested, 3%, or three thousand, actually do have those antibodies, but 97%, or ninety-seven thousand, do not (white boxes, first column with data in it).

Of the 3,000 folks who really do have the antibodies – first line of data – we have a false  negative rate of 10%, so three hundred of these poor folks are given the false good tidings that they have never been exposed (that’s the upper orange box). The other 90% of them, or two thousand seven hundred, are told, correctly, that they have been exposed (that’s the upper green box).

Now of the 97,000 people who really do NOT have any antibodies – the second line of data – we have a false positive rate of 7%, so you multiply 0.07 times 97000 to get six thousand, seven hundred ninety of them who would be told, incorrectly, that they DID test positive for Covid-19 – in the lower orange box. (Remember, positive is bad here, and negative is good.) However, 90,210 would be told, correctly, that they did not have those antibodies. (That’s in the lower green box.)

Now let’s add up the folks who got the positive test results, which is the third data column. We had 2,700 who correctly tested positive and 6,790 who wrongly tested positive. That’s a total of 9,490 people with a positive CV19 antibody test, which means that of that group of people, only 28.5% were correctly so informed!! That’s between a third and a fourth! Unacceptable!

However, if we look at the last column, notice that almost every single person who was told that they were negative, really was negative. (Donno about you, but I think that 99.7% accuracy is pretty darned good!)

However, that 28.5% accuracy among the ‘positives’ (in the left-hand blue box) is really worrisome. What to do?

Simple! Test those folks again! Right away! Let’s do it, and then let’s look at the results:

math of CV19 testing - round 2

Wowser! We took the 9490 people who tested positive and gave them another round of tests, using the exact same equipment and protocols and error rates as the first one. The spreadsheet is set up the same; the only thing I changed is the bottom two numbers in the first data column. I’m not going to go through all the steps, but feel free to check my arithmetic. Actually, check my logic. Excel doesn’t really make arithmetic errors, but if I set up the spreadsheet incorrectly, it will spit out incorrect results.

Notice that our error rate (in blue) is much lower in terms of those who tested positive. In fact, of those who test positive, 83.7% really ARE positive this time around, and of those who test negative, 95.9% really ARE negative.

But 84% isn’t accurate enough for me (it’s either a B or a C in most American schools). So what do we do? Test again – all of the nearly three thousand who tested positive the first time. Ignore the rest.

Let’s do it:

math of CV19 testing - round 3

At this point, we have much higher confidence, 98.5% (in blue), that the people who tested ‘positive’, really are ‘positive’. Unfortunately, at this point, of the people who tested negative, only about 64% of the time is that correct. 243 people who really have the antibodies tested negative. So perhaps one should test that subgroup again.

The beautiful thing about this method is that it doesn’t even require a terribly exact test! But it does require that you do it repeatedly, and quickly.

Let me assure you that the exact level of accuracy, and the exact number of exposed people, doesn’t matter: If you test and re-test, you can find those who are infected with almost 100% accuracy. With that information you can then discover what the best approaches are to solving this pandemic, what the morbidity and mortality rates are, and eventually to stop it completely.

Why we don’t have enough tests to do this quickly and accurately and repeatedly is a question that I will leave to my readers.

Addendum:

Note that I made some starting assumptions. Let us change them and see what happens. Let’s suppose that the correct percentage of people with COVID-19 antibodies is not 3%, but 8%. Or maybe only 1%. Let’s also assume a 7% false positive and a 10% false negative rate. How would these results change? With a spreadsheet, that’s easy. First, let me start with an 8% infection rate and keep testing repeatedly. Here are the final results:

Round Positive accuracy rating Negative accuracy rating
1 52.8% 99.1%
2 93.5% 89.3%
3 99.5% 39.3%

So after 3 rounds, we have 99.5% accuracy.

Let’s start over with a population where only 1% has the antibodies, and the false positive rate is 7% and the false negative rate is 10%.

Round Positive accuracy rating Negative accuracy rating
1 11.5% 99.9%
2 62.6% 98.6%
3 95.6% 84.7%
4 99.6% 30.0%

This time, it took four rounds, but we still got to over 99.6% accuracy at distinguishing those who really had been exposed to this virus. Yes, towards the end our false negative rate rises, but I submit that doesn’t matter that much.

So Parson Tommy Bayes was right.

More on the “false positive” COVID-19 testing problem

I used my cell phone last night to go into the problem of faulty testing for COVID-19, based on a NYT article. As a result, I couldn’t make any nice tables. Let me remedy that and also look at a few more assumptions.

This table summarizes the testing results on a theoretical group of a million Americans tested, assuming that 5% of the population actually has coronavirus antibodies, and that the tests being given have a false negative rate of 10% and a false positive rate of 3%. Reminder: a ‘false negative’ result means that you are told that you don’t have any coronavirus antibodies but you actually do have them, and a ‘false positive’ result means that you are told that you DO have those antibodies, but you really do NOT. I have tried to highlight the numbers of people who get incorrect results in the color red.

Table A

Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 50,000 10% 45,000 5,000
Actually Negative 950,000 3% 28,500 921,500
Totals 1,000,000 73,500 926,500
Percent we assume are actually positive 5% Accuracy Rating 61.2% 99.5%

As you can see, using those assumptions, if you get a lab test result that says you are positive, that will only be correct in about 61% of the time. Which means that you need to take another test, or perhaps two more tests, to see whether they agree.

The next table assumes again a true 5% positive result for the population and a false negative rate of 10%, but a false positive rate of 14%.

Table B

Assume 5% really exposed, 14% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 50,000 10% 45,000 5,000
Actually Negative 950,000 14% 133,000 817,000
Totals 1,000,000 178,000 822,000
Percent we assume are actually positive 5% Accuracy Rating 25.3% 99.4%

Note that in this scenario, if you get a test result that says you are positive, that is only going to be correct one-quarter of the time (25.3%)! That is useless!

Now, let’s assume a lower percentage of the population actually has the COVID-19 antibodies, say, two percent. Here are the results if we assume a 3% false positive rate:

Table C

Assume 2% really exposed, 3% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 20,000 10% 18,000 2,000
Actually Negative 980,000 3% 29,400 950,600
Totals 1,000,000 47,400 952,600
Percent we assume are actually positive 2% Accuracy Rating 38.0% 99.8%

Notice that in this scenario, if you get a ‘positive’ result, it is likely to be correct only a little better than one-third of the time (38.0%).

And now let’s assume 2% actual exposure, 14% false positive, 10% false negative:

Table D

Assume 2% really exposed, 14% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 20,000 10% 45,000 2,000
Actually Negative 980,000 14% 137,200 842,800
Totals 1,000,000 182,200 844,800
Percent we assume are actually positive 2% Accuracy Rating 24.7% 99.8%

Once again, the chances of a ‘positive’ test result being accurate is only about one in four (24.7%), which means that this level of accuracy is not going to be useful to the public at large.

Final set of assumptions: 3% actual positive rate, and excellent tests with only 3% false positive and false negative rates:

Table E

Assume 3% really exposed, 3% false positive rate, 3% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 30,000 3% 45,000 900
Actually Negative 970,000 3% 29,100 940,900
Totals 1,000,000 74,100 941,800
Percent we assume are actually positive 3% Accuracy Rating 60.7% 99.9%

Once again, if you test positive in this scenario, that result is only going to be correct about 3/5 of the time (60.7%).

All is not lost, however. Suppose we re-test all the people who tested positive in this last group (that’s a bit over seventy-four thousand people, in Table E). Here are the results:

Table F

Assume 60.7% really exposed, 3% false positive rate, 3% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 45,000 3% 43,650 1,350
Actually Negative 29,100 3% 873 28,227
Totals 74,100 44,523 29,577
Percent we assume are actually positive 60.7% Accuracy Rating 98.0% 95.4%

Notice that 98% accuracy rating for positive results! Much better!

What about our earlier scenario, in table B, with a 5% overall exposure rating, 14% false positives, and 10% false negatives — what if we re-test all the folks who tested positive? Here are the results:

Table G

Assume 25.3% really exposed, 14% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 45,000 14% 38,700 6,300
Actually Negative 133,000 10% 13,300 119,700
Totals 178,000 52,000 126,000
Percent we assume are really positive 25.3% Accuracy Rating 74.4% 95.0%

This is still not very good: the re-test is going to be accurate only about three-quarters of the time (74.4%) that it says you really have been exposed, and would only clear you 95% of the time. So we would need to run yet another test on those who again tested positive in Table G. If we do it, the results are here:

Table H

Assume 74.4% really exposed, 14% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 38,700 14% 33,282 5,418
Actually Negative 13,300 10% 1,330 11,970
Totals 52,000 34,612 17,388
Percent we assume are really positive 74.4% Accuracy Rating 96.2% 68.8%

This result is much better, but note that this requires THREE TESTS on each of these supposedly positive people to see if they are in fact positive. It also means that if they get a ‘negative’ result, that’s likely to be correct only about 2/3 of the time (68.8%).

So, no wonder that a lot of the testing results we are seeing are difficult to interpret! This is why science requires repeated measurements to separate the truth from fiction! And it also explains some of the snafus committed by our current federal leadership in insisting on not using tests offered from abroad.

 

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EDIT at 10:30 pm on 4/25/2020: I found a few minor mistakes and corrected them, and tried to format things more clearly.

COVID-19 Numbers in the US do not seem to be growing exponentially

Looking at the past month of CDC-reported infections and deaths from the new corona virus, I conclude that there has been some good news: the total number of infections and deaths are no longer following an exponential growth curve.

The numbers are indeed growing, by either a quadratic (that is, x^2) or a quartic (x^4) curve, which is not good, and there is no sign of numbers decreasing.

BUT it looks as though the physical-social distancing and self-quarantining that I see going on around me is actually having an effect.

Yippee!

Here is my evidence: the actual numbers of infected people are in blue, and the best-fit exponential-growth equation is in red. You can see that they do not match well at all. 

total cases US not looking exponential

If they did match, and if this were in fact exponential growth, we would have just about the entire US population infected by the end of just this month of April – over 300 million! That no longer seems likely. Take a look at the next graph instead, which uses the same data, but polynomial growth:

total cases US looking second power

Just by eyeballing this, you can see that the red dots and blue dots match really, really well. When I extend the graph until the end of April, I get a predicted number of ‘only’ 1.5 million infected. Not good, but a whole lot better than the entire US population!

Also, let’s look at total cumulative reported deaths so far. Here are the CDC-reported numbers plotted against a best-fit exponential curve:

deaths do not seem to be exponential

Up until just a few days ago, this graph was conforming pretty well to exponential growth. However, since about April 8, that seems to be no longer the case. If the total numbers of deaths were in fact growing at the same percentage rate each day, which is the definition of exponential growth, then by the end of April we would have 1.5 million DEAD. That’s THIS MONTH. Continued exponential growth would have 1.2 BILLION dead in this country alone by the end of May.

Fortunately, that is of course impossible.

Unfortunately all that means is that the virus would run out of people to infect and kill, and we would get logistic growth (which is the very last graph, at the bottom).

death seem to be 4th power polynomial

This fourth-degree mathematical model seems to me to work much better at describing the numbers of deaths so far, and has a fairly good chance of predicting what may be coming up in the near future. It’s still not a good situation, but it shows to me that the social and physical distancing we are doing is having a positive effect.

But let’s not get complacent: if this model correctly predicts the next month or two, then by the end of April, we would have about 60 thousand dead, and by the end of May we would have 180 thousand dead.

But both of those grim numbers are much, much lower than we would have if we were not doing this self-isolation, and if the numbers continued to grow exponentially.

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FYI, a logistic curve is shown below. Bacteria or fungi growing in a broth will grow exponentially at first, but after a while, they not only run out of fresh broth to eat, but they also start fouling their own environment with their own wastes. WE DO NOT WANT THIS SITUATION TO HAPPEN WITH US, NAMELY, THAT WE ALL GET INFECTED!!!

logistic curve again

Various graphs for deaths from COVID-19, so far

I wrote that I would show you what various graphs of various types of simple models look like for deaths so far due to the current corona virus: linear, exponential, polynomial, and so on. I think that a fourth-degree (not third-degree, like I wrote earlier) seems to fit the data best so far, and that’s better than exponential growth.

First, let’s look at a straight-line best-fit model, superimposed by Excel on the data. (Note: deaths are on the Y, or vertical, axis; the X-axis represents days since the beginning of March. So today, the 6th of April, is day 37 (31 + 6). The dotted red line represents the line of best fit, and the blue dots are the CDC-announced numbers of deaths so far.

is it linear

As you can see, the straight dotted line doesn’t fit the data very well at all. R-squared, known as the correlation coefficient, tells us numerically how well it fits. If R or R^2 equals 1.000, then you have absolutely perfect correlation of the data to your model. Which we do NOT have here. By the way, in that model, then by mid-June we would have about 22,000 dead from this disease.

OK, let’s look at an exponential curve-of-best fit next:

is it exponential

As you can see, this red curve fits the data a LOT better, and R-squared is a lot higher.

Unfortunately.

We do NOT WANT EXPONENTIAL GROWTH OF THIS OR ANY OTHER DISEASE, BECAUSE IT MEANS WE ALL GET IT! In fact, if this model is accurate and isn’t slowed down, then by mid-June, just plugging in the numbers, we would have 3.3 BILLION (not million) people dead in the US alone. Fortunately, that won’t happen.

BUT there are some parts of the data where the curve doesn’t fit perfectly — let me point them out:

is it exponential -2

At the upper right-hand end, the red dotted line is quite a bit higher than the blue dots. Fortunately. And near the middle of the graph, the blue dots of death are higher than the red line.

OK, let’s look at some polynomial models instead:

is it a second degree polynomial

This is a fancy version of the simple y=x^2 parabolas you may have graphed in Algebra 1. Once again, this doesn’t do a terrific job of conforming to the actual data. At the right-hand end, the blue dots of death are higher than the curve. In addition, if we continued the red curve to the left, we would find that something like two thousand people had already died in the US, and presumably came back to life. Which is ridiculous.

However, if this model were to hold true until mid-June, we would have 127 thousand dead. Not good.

Let’s try a third-degree polynomial (a cubic):

is it a third degree polynomial

That’s pretty remarkable agreement between the data and the equation! That’s the equation I was using in my earlier post. The R-squared correlation is amazing. Unfortunately, if this continues to hold, then we would have about 468 thousand dead in the US.

Let’s continue by looking at a fourth-degree polynomial curve fitted to the data:

is it a fourth degree polynomial

That is an amazingly good fit to the data! Unfortunately, let’s hope that it won’t continue to fit the data, because if it does, then we are looking at a little over a MILLION dead.

Let’s hope we can get these totals to level off by physically distancing ourselves from other households, washing our hands, and getting proper protective garments and testing technology to our medical personnel.

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Here’s another model that unfortunately does NOT work: logarithmic growth. If it were the case, then we would have about 10,700 deaths by mid-June.

is it logarithmic

 

People are Not Cattle!

This apparently did not occur to William Sanders.

He thought that statistical methods that are useful with farm animals could also be used to measure effectiveness of teachers.

I grew up on a farm, and as both a kid and a young man I had considerable experience handling cows, chickens, and sheep. (These are generic critter photos, not the actual animals we had.)

I also taught math and some science to kids like the ones shown below for over 30 years.

guy teaching  deal students

Caring for farm animals and teaching young people are not the same thing.

(Duh.)

As the saying goes: “Teaching isn’t rocket science. It’s much harder.”

I am quite sure that with careful measurements of different types of feed, medications, pasturage, and bedding, it is quite possible to figure out which mix of those elements might help or hinder the production of milk and cream from dairy cows. That’s because dairy or meat cattle (or chickens, or sheep, or pigs) are pretty simple creatures: all a farmer wants is for them to produce lots of high-quality milk, meat, wool, or eggs for the least cost to the farmer, and without getting in trouble.

William Sanders was well-known for his statistical work with dairy cows. His step into hubris and nuttiness was to translate this sort of mathematics to little humans. From Wikipedia:

“The model has prompted numerous federal lawsuits charging that the evaluation system, which is now tied to teacher pay and tenure in Tennessee, doesn’t take into account student-level variables such as growing up in poverty. In 2014, the American Statistical Association called its validity into question, and other critics have said TVAAS should not be the sole tool used to judge teachers.”

But there are several problems with this.

  • We  don’t have an easily-defined and nationally-agreed upon goal for education that we can actually measure. If you don’t believe this, try asking a random set of people what they think should be primary the goal of education, and listen to all the different ideas!
  • It’s certainly not just ‘higher test scores’ — the math whizzes who brought us “collateralization of debt-swap obligations in leveraged financings” surely had exceedingly high math test scores, but I submit that their character education (as in, ‘not defrauding the public’) was lacking. In their selfishness and hubris, they have succeeded in nearly bankrupting the world economy while buying themselves multiple mansions and yachts, yet causing misery to billions living in slums around the world and millions here in the US who lost their homes and are now sleeping in their cars.
  • Is our goal also to ‘educate’ our future generations for the lowest cost? Given the prices for the best private schools and private tutors, it is clear that the wealthy believe that THEIR children should be afforded excellent educations that include very small classes, sports, drama, music, free play and exploration, foreign languages, writing, literature, a deep understanding and competency in mathematics & all of the sciences, as well as a solid grounding in the social sciences (including history, civics, and character education). Those parents realize that a good education is expensive, so they ‘throw money at the problem’. Unfortunately, the wealthy don’t want to do the same for the children of the poor.
  • Reducing the goals of education to just a student’s scores on secretive tests in just two subjects, and claiming that it’s possible to tease out the effectiveness of ANY teacher, even those who teach neither English/Language Arts or Math, is madness.
  • Why? Study after study (not by Sanders, of course) has shown that the actual influence of any given teacher on a student is only from 1% of 14% of test scores. By far the greatest influence is from the student’s own family background, not the ability of a single teacher to raise test scores in April. (An effect which I have shown is chimerical — the effect one year is mostly likely completely different the next year!)
  • By comparison, a cow’s life is pretty simple. They eat whatever they are given (be that straw, shredded newspaper, cotton seeds, chicken poop mixed with sawdust, or even the dregs from squeezing out orange juice [no, I’m not making that up.]. Cows also poop, drink, pee, chew their cud, and sometimes they try to bully each other. If it’s a dairy cow, it gets milked twice a day, every day, at set times. If it’s a steer, he/it mostly sits around and eats (and poops and pees) until it’s time to send  them off to the slaughterhouse. That’s pretty much it.
  • Gary Rubinstein and I have dissected the value-added scores for New York City public school teachers that were computed and released by the New York Times. We both found that for any given teacher who taught the same subject matter and grade level in the very same school over the period of the NYT data, there was almost NO CORRELATION between their scores for one year to the next.
  • We also showed that teachers who were given scores in both math and reading (say, elementary teachers), there was almost no correlation between their scores in math and in reading.
  • Furthermore, with teachers who were given scores in a single subject (say, math) but at different grade levels (say, 6th and 7th grade math), you guessed it: extremely low correlation.
  • In other words, it seemed to act like a very, very expensive and complicated random-number generator.
  • People have much, much more complicated inputs, and much more complicated outputs. Someone should have written on William Sanders’ tombstone the phrase “People are not cattle.”

Interesting fact: Jason Kamras was considered to be the architect of Value-Added measurement for teachers in Washington, DC, implemented under the notorious and now-disgraced Michelle Rhee. However, when he left DC to become head of Richmond VA public schools, he did not bring it with him.

 

A geometry lesson inspired by a silvering company – and a rant about computerized learning programs

Here is some information that teachers at quite a few different levels could use* for a really interesting geometry lesson involving reflections involving two or more mirrors, placed at various angles!

Certain specific angles have very special effects, including 90, 72, 60, 45 degrees … But WHY?

This could be done with actual mirrors and a protractor, or with geometry software like Geometer’s Sketchpad or Desmos. Students could also end up making their own kaleidoscopes – either with little bits of colored plastic at the end or else with some sort of a wide-angle lens. (You can find many easy directions online for doing just that; some kits are a lot more optically perfect than others, but I don’t think I’ve even seen a kaleidoscope that had its mirrors set at any angle other than 60 degrees!)

I am reproducing a couple of the images and text that Angel Gilding provides on their website (which they set up to sell silvering kits (about which I’ve posted before, and which I am going to attempt using pretty soon, on a large astronomical mirror I’ve been polishing for quite some time)).

At 72º you see 4 complete reflections.

When two mirrors are parallel to each other, the number of reflections is infinite. Placing one mirror at a slight angle causes the reflections to curve.

 

https://angelgilding.com/multiple-reflections/

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Rant, in the form of a long footnote:

* assuming that the teacher are still allowed to initiate and carry out interesting projects for their students to use, and aren’t forced to follow a scripted curriculum. It would be a lot better use of computers than forcing kids to painfully walk through (and cheat, and goof off a lot) when an entire class is forced to use one of those very expensive but basically worthless highly-centralized, district-purchased computer-managed-instruction apps. God, what a waste of time – from personal experience attempting to be a volunteer community math tutor at such a school, and also from my experience as a paid or volunteer tutor in helping many many students who have had to use such programs as homework. Also when I was required to use them in my own classes, over a decade ago, I and most of my colleagues found them a waste of time. (Not all – I got officially reprimanded for telling my department chair that ‘Renaissance Math’ was either a ‘pile of crap’ or a ‘pile of shit’ to my then-department head, in the hearing of one of the APs, on a teacher-only day.

Keep in mind: I’m no Luddite! I realized early on that in math, science, and art, computers would be very, very useful. I learned how to write programs in BASIC on one of the very first time-share networks, 45 years ago. For the first ten years that my school system there was almost no decent useful software for math teachers to use with their classes unless you had AppleII computers. We had Commodore-64’s which were totally incompatible and there were very few companies (Sunburst was one) putting out any decent software for the latter. So when I saw some great ideas that would be ideal for kids to use on computers to make thinking about numbers, graphs, and equations actually fun and mentally engaging, often I would have to write them my self during whatever free time I could catch, at nights and weekends. Of course, doing this while being a daddy to 2 kids, and still trying to teach JHS math to a full load of students (100 to 150 different kids a day at Francis Junior High School) and running a school math club and later coaching soccer. (I won’t say I was a perfect person or a perfect teacher. I believe I learned to give better math explanations than most, didn’t believe that you either have a ‘m,ath gene’ or you don’t, at times had some interesting projects, and at times was very patient and clear, but had a terrible temper and often not good at defusing things. Ask my kids or my former students!) Later on, I collaborated with some French math teachers and a computer programmer to try to make an app/program called Geometrix for American geometry classes that was supposed to help kids figure out how to make all sorts of geometric constructions and then develop a proof of some property of that situation. It was a failure. I was the one writing the American version, including constructions and tasks from the text I was currently using. There was no way I could anticipate what sorts of obstacles students would find when using this program, until I had actual guinea pig students to use them with. Turns out the final crunch of writing however many hundreds of exercises took place over the summer, and no students to try them on. Figuring out hints and clues would require watching a whole bunch of kids and seeing what they were getting right or wrong. In other words, a lot of people’s full time job for a long time, maybe paying the kids as well to try it out so as to get good feedback, and so on. Maybe it could work, but it would require a lot more investment of resources that the tiny French and American companies involved could afford. We would have really needed a team of people, not just me and a single checker.

I find that none of these computer-dominated online learning programs (much less the one I worked on) can take the place of a good teacher. Being in class, listening to and communicating logically or emotionally with a number of other students and a knowledgeable adult or two, is in itself an extremely important skill  to learn. It’s also the best way to absorb new material in a way that will make sense and be added to one’s store of knowledge. That sort of group interaction is simply IMPOSSIBLE in a class where everybody is completely atomized and is on their own electronic device, engaged or not.

Without a human being trying to make sense out of the material, what I found quite consistently, in all the computerized settings, that most students absorbed nothing at all or else the wrong lessons altogether (such as, ‘if you randomly try all the multiple choice answers, you’ll eventually pick the right one and you can move on to some other stupid screen’; it doesn’t matter that all your prior choices were wrong; sometimes you get lucky and pick the right one first or second! Whee! It’s like a slot machine at a casino!).

By contrast, I found that with programs/apps/languages like Logo, Darts, Green Globs, or Geometer’s Sketchpad, with teacher guidance, students actually got engaged in the process, had fun, and learned something.

I find the canned computer “explanations” are almost always ignored by the students, and are sometimes flat-out wrong. Other times, although they may be mathematically correct, they assume either way too much or way too little, or else are just plain confusing. I have yet to detect much of any learning going on because of those programs.

 

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