Anne Stone and Jeff Nichols beautifully make the case against standardized testing at the New York Times SchoolBook blog (hat-tip to Susan Ohanian). Here is an excerpt:

In light of current controversies around testing and teacher evaluation, let’s do a little thought experiment. How would Miss Snug have handled this lesson if it were occurring just before a round of standardized testing? Would she not have had to interrupt the children’s speculations and instructed them that actual circumstances in word problems must be completely disregarded, because the point is to arrive at the answer the test designers have in mind? After all, how could test designers anticipate the lines of thought that spontaneously erupted in her classroom? Real life, and real thought, are too complicated to be foreseen – and so need to be put aside at testing time.

We … are concerned about education because our adult citizens need to be flexible thinkers, ready to adapt to the ever-changing circumstances of the global marketplace. But making standardized tests the center of our curriculums tells children the most important thing they need to learn in school is how to arrive at predetermined answers on the tests. …

So in fact the test doesn’t reflect at all what kids should learn in school. What they really need to master is the kind of imaginative, adaptive thinking Miss Snug encourages in the passage from “The Trumpet of the Swan” – skills that cannot be assessed in any way other than actually knowing the children.

This little episode captures what volumes of education research have shown: we are born curious, and the best education models do not proceed on the basis of “what we want them to learn,” as Mr. Bloomberg correctly describes the goal of test-oriented education, but on the assumption that our job is to foster children’s ability ultimately to shape a world different from what we leave to them.

Now I agree that basic skills in mathematics are important. To learn mathematics effectively, you need to have your multiplication facts at your fingertips, and also your trig identities (if you’re learning at a higher level). However, the problem is that our assessment systems are skewed towards what is easy to test, especially what is easy to test using standardized tests. When resources are strained, as they are now, what is easy to assess ends up dominating the entire educational enterprise.

The best way to engage students is to ask them what they wish to learn. What is it that makes their hearts sing? (However, we should begin asking them early in life; if you wait until 12 or 16 years of the standard education system has crushed the spirit of our children, then you might not get a very satisfying answer to what makes their hearts sing.) And then create an environment where they are lovingly supported in pursuing their own goals. This is the only way to nurture creative and critical problem-solvers that we need so much to confront the serious problems facing civilization. Assessing their work can never be done using standardized tests.

In pursuing their own goals, many students will have need for some mathematical technique or other. With 21st century technology, it should be possible to create a system, a repository, of lessons that teach all of the basic techniques. (I’m sure that the internet is already chock full of such material, albeit scattered and of uneven quality.) When students need them, they can be guided and supported in learning what they need. Making students sit through 12 years of mathematics through high school, and then even more afterwards, in case they should need it, is a terrible waste of effort. Students would be better off *doing something* of value.

I cringe every time I think about the excellent teachers out there spending precious class time teaching 500 students how to work the product rule for derivatives and such things. Such things are best learned by doing, with feedback, in a combination of small groups and alone. I would much rather see our excellent lecturers spending precious class time on modelling the kind of critical and creative thinking that can’t easily be transmitted in other ways.

A final note: The inhumane standardized testing that is consuming the education landscape like a plague is infecting teacher evaluation as well. Tim Clifford’s article at the same SchoolBook blog is a powerful argument against this horrible practice.

It’s always a risk letting go and allowing students to discover a concept for themselves. Part of the problem is the reality that many high school kids’ ‘curiosity’ does not extend to determining the number if zeros of a quadratic from standard form (for example). Many lack the interest or discipline to see a problem through to the end. If there is no effective consolidation at the end of class to ensure they have ‘arrived’ then the class time was truly wasted. Perhaps, as the article mentioned, starting young is the key. Still, not everyone’s natural curiosities bend in the mathematical direction, yet all find themselves taking three credits minimum of high school math. It’s a growing ability for me to structure exploration activities to maximize focus and effectiveness. I still believe discovery is better than lecture learning, but there is the other side of the coin where we shouldn’t reinvent the wheel. Math is tapestry of thousands of years across many cultures; it is idealistic to presume even the most curious would stumble down all the necessary roads in the short span of the public school years.

Nice to hear from you, Tim!

I am not arguing that we should be reinventing the wheel every day. I am arguing that the structure of the education system ought to change. Right now the education system forces excellent teachers such as you to “cover” a certain curriculum every year, whether it is of interest to students or not. But really, is it essential for *everyone* to learn the quadratic formula? My wife has built a truly wonderful career, helps an enormous number of people, and never gives the quadratic formula a thought. I once asked a chemical engineer how often he uses calculus; his answer was, “Never. I just look up numbers in tables when I need them.” We persist in teaching university math and science as if all math and science majors will become math or science professors, which is not very effective, since only a minuscule number of them will do so.

Our curricula are one-size-fits-all, and they are content-oriented. What I am suggesting is that teachers should work with students to decide what would be best for students to learn, based on the interests of their current students. There ought to be resources available so that when the need for technical skills arise, students have a place to go to learn what they need. Students should pursue studies of their own interest, and if they involve math, then they should learn the math as the need arises.

I don’t think it is of much value to force students to sit through high-school math just in case they might go into engineering, or whatever. We keep telling students to learn the material, that they will see the use of it later. But for almost all students, later never comes. So math becomes a drag for almost everyone. (And it is no better at university; it gets worse there, because courses are so fast-paced, and so packed with content, that there is no time for most students to see any beauty in the subject, and there is no time for most profs to expose any of the beauty.) I would rather see them pursue their own interests, guided by nurturing teachers. But the system does not allow for this; unfortunately, a central authority decides what you, Tim Calford, must do in your class. This, to me, is a real pity. I would much rather see you have the freedom to exercise your own creativity in stimulating your students to learn what is their heart’s desires.

We are afraid to do this, because we think that nobody will learn anything and a lot of time and resources will be wasted. But ask any adult now what they remember of their high-school math classes, and what valuable skills they learned in high-school math classes that now benefit their lives or inform their careers. To me, the system as it is constructed wastes the time of most students and wastes the enthusiasm of our many wonderful teachers.

We need to have enough courage and confidence in our excellent teachers to allow them the freedom to provide a truly inspiring education for their students, one that will teach them critical and creative thinking. But powerful forces are against this; we would rather *standardize* teaching and testing. We will pay the price.

But I hope that I haven’t dampened your enthusiasm for teaching; we need wonderful teachers like you to do the best they can under the circumstances, until we can change circumstances for the better, for both students and teachers.

Saludos from the Caribbean! Professor, I have a geometry question for you. I’m teaching the SAT to high school students here, and I’ve come across a problem that is making me insane. 😛 Actually, it’s not on the SAT; it’s a problem from the ASHME. If you can help me solve it, I would greatly appreciate it. I’ve spent two days looking at this and last night I dreamed about it. This is a good sign because it means I’m getting close to the answer.

I don’t have a scanner but I will describe the problem. It’s simple geometry…but with a twist of the morbid. 🙂

You have trapezoid ABCD inscribed a circle. The trapezoid’s longest base also happens to be the circle’s diameter, which is CD. (Hence, the smaller base is AB). If the radius of the circle with center M is 10, and AC is congruent to BD, and you have a radius extending from M to point F at a perpendicular to CD, then what is the length of EF? (E being the point where the radius cuts the trapezoid in half.

I hope that made sense. Basically, AB is not only the trapezoid’s smallest base but also a chord of the circle. What the problem is asking is: how long is that little tiny piece extending from E to F. We know that MF measures 10 because that’s the radius. So to rephrase: what is the length from M to E and from E to F? These two lengths should add up to 10.

I wish I had a scanner. Thank you! Multi grazie, como siempre.

Oh wait, I forgot to tell you. I think the length of EF is 4, but I’m not sure if I’m right.

Saludos, Pola!

A good strategy to pass on to your students (for dealing with such geometry problems) is to imagine moving the points and lines around. I learned this lesson in my very first year of teaching, when a student asked me if I could show the proof of a theorem from geometry. He then stated the theorem and I dug in, thinking it should be quick and easy. After some time with no success, I suddenly said, “Wait … is it even correct?” It took very little time to determine (by imagining the line segments to be rigid sticks that could be moved around) that the theorem was in fact not correct, and I learned a good lesson.

There is a software program called Geogebra that is both free and excellent, and might be useful to your students, if they don’t already know it. One can create the kinds of geometrical figures that you described, and then play with them. When you “grab” a point and move it, the connected line segments also move, and so the software allows you to create a kind of playground where a student can gain experience and develop intuition.

OK, back to your problem. If I understand you correctly, AB is parallel to CD, F is on the circle, MF is perpendicular to AB, and E is the intersection point of AB and MF. If this is so, there is not enough information to solve the problem. You can see this by sliding the line segment AB so that it remains parallel to CD, but its distance from CD changes. No matter the position of AB (and the position of E, and therefore the answer to the question, depends on the position of AB) the conditions of the problem are still satisfied.

Is there any additional information that you forgot to send me? Or am I misinterpreting the problem?

Adios,

Santo

ps. I have read your blog with appreciation. You write very well, engaging your reader with posts that are interesting and sometimes very funny!

Oh wow, that is SUCH a great idea! I had not thought of that! I need to move those points around. That’s going to make geometry SO MUCH BETTER for my students, and for me as well. 🙂 It’s my first year teaching this course, and I’m constantly learning. Thank you!!!

Geogebra: I am going to download that RIGHT NOW and recommend it to my students. I checked out its Web site, and I loved it.

I can’t believe you actually understood what I wrote! After I hit “Post Comment” I thought, Wow…that made no sense at all. So yes, you understood it completely. I contacted the teacher who assigned the problem in class and she said: “Oh! I forgot! There’s a line from A to M and another line from B to M. If you look at these lines, they’re actually radii, measuring 10 each. And AC and BD are congruent!”

So I went back and plugged this in…but was still unable to arrive at an answer. I think I’ve reached a supersaturation point with this problem. I have officially arrived at “analysis paralysis.”

I’m downloading Geogebra as we speak! Thank you for responding to my post so quickly. I really appreciate it!!! And thank you for reading my blog…it’s a work in progress! 🙂 Ciao! Sofía 🙂

Hola Sofia,

If I understand the diagram correctly, then even with the additional information provided, there is still not enough information to solve the problem. This must be very frustrating for you!

I’ll send a further suggestion by email.

All the best,

Santo

I knew it!!! I knew it!!! Thank you! The extra information doesn’t change anything about the problem. Saludos! Sofía

Thanks for the thoughtful response Santo! I don’t want to seem rude by my delayed reply to your reply… I’ve just been busy this week. I’ll get back to you this weekend.. I was going to write, but the read the geometry chat going on and got distracted constructing the circle and trapezoid! I came to the same conclusion, that the conditions do necessitate a unique solution, since any position for point A and B would have a radius of 10.

Hi Tim,

No worries, we can continue the discussion any time. It’s always nice to hear from you.

Take care,

Santo

Hello, hello! Good Sunday! Buen domingo! A student was smart and decided to send me an official copy of the problem:

http://goldenvalley.kernhigh.org/robledo/Classes/geometry/ch09/section02/MASTERSB.pdf

It’s problem #4. The answer is 2. Somehow, you have to assume that triangle MEB and CAM are 6-8-10 triangles. If that’s the case, then FE is 2.

My goodness, I may have to ask an Italian friend to bring in the sambuca (sambucca?). Thank you!

Buon giorno Sofia,

Compare problems 3 and 4, where two values are given. In each of these problems one can use Pythagoras’s theorem and the fact that triangle MEB is a right triangle to determine EM. However, in problem 2 only one value is given … not enough information. Sure, if you make additional assumptions then you can solve the problem, but these additional assumptions need to be specified in the problem. If you assume that triangle MEB is a 6-8-10 triangle, then ME = 8, and FE = 10 – 8 = 2. I doubt that this is what was intended, but who knows?

Ciao,

Santo

Santo,

I was in one of your Physics classes a few years back. Considering Physics was a context credit for my major you still made it ridiculously enjoyable. You were one of my favorite professors at Brock and I just wanted to display my thanks for the time that I had you as my professor.

Thanks very much for your very kind words!

All of science is based on testing, all of which includes some form of complexity reduction and/or simplification. A model of a house in a wind tunnel isn’t really a house in a storm. But we accept the problems this sort of standardized testing entails, and simply point out the error bars.

Of, but god forbid someone should attempt to apply the same to kids. In that case, it’s “inhuman”.

Sure, I can think of all sorts of problems with standardized testing. But I can think of all sorts of problems with wind tunnel testing too. But when we’re talking about wind tunnels, we don’t give up, we immediately attack these problems looking for ways to improve it. But in this case, the argument is invariably that it not only can’t work, but is inherently evil to even try.

Is there an *actual* problem here? How would you demonstrate that? It wouldn’t be by some sort of measured result, would it?

Of, but god forbid someone should attempt to apply the same to kids. In that case, it’s “inhuman”.I presume you are being sarcastic, but I do think the *extreme* overemphasis on high-stakes standardaized testing is inhumane. But beyond inhumane, it is also extremely ineffective in fostering good teaching and learning. I have written about this in a number of my blog posts, so I won’t repeat the arguments here, but refer you to the relevant posts if you are interested.

You talk about wind tunnels, and how all of science is based on testing, but learning and teaching is more art than science. Students are not electrons, that can be tested by running them through an apparatus over and over again.

It boils down to what your goals for the education system are, keeping in mind that citizens are paying for the system. If your goal is to have a system where you don’t care about most students, as long as you have a sufficient number of students that reach high levels of achievement, then there is probably no problem. But in this case, why have a system in place at all? We citizens could save a lot of money by scrapping the entire system, because high achievers will achieve no matter what education system is used, even if there is none. I would prefer a goal along the lines of producing a large number of highly educated citizens, with an education system designed to nurture students by providing a loving, supportive environment where they could follow their own learning desires. I believe we could do a lot better than the current system in producing many more students that are self-motivated, highly creative, and with a much higher level of critical thinking skills, of which the world has abundant need.

Is there an *actual* problem here? How would you demonstrate that? It wouldn’t be by some sort of measured result, would it?Again, it depends on what your goals for our education system are. If you just want to keep students off the streets, then no, there is probably no problem. If you want the system to produce graduates that are operating on a high level in some reasonable ways, then there is lots of measured evidence that we have a big problem. Lots of high-school graduates cannot read very well, cannot write very well, have virtually no sense for science, cannot think very well; and all this for a very large outlay at taxpayer’s expense. Now ask yourself in what way will high-stakes standardized testing help students do any of this better? There are some very rich and very powerful people, who know nothing about education and have never been teachers, who reckon that they have the solution that will fix what they see are problems in the education system. Some of these solutions involve greatly expanding the scope of standardized tests, and even tying teacher evaluations to standardized tests. All of this has the effect of placing a great deal of pressure on teachers and students to focus a large amount of time and effort to filling in blanks and selecting which of four alternatives is the best response to a question. But think about it: Consider some “real world” skill that one might like an employee to have. Let’s say you want one of your employees to learn how to write well, or learn how to be a good sales person, or how to better fix machines on the assembly line, or how to respond more effectively to clients having software problems. I can’t think of any situation where using standardized tests would be a better training tool than getting the employee to actually do whatever the task is, watch them do the task, and provide them with personal feedback.

Sure, I can think of all sorts of problems with standardized testing. But I can think of all sorts of problems with wind tunnel testing too. But when we’re talking about wind tunnels, we don’t give up, we immediately attack these problems looking for ways to improve it. But in this case, the argument is invariably that it not only can’t work, but is inherently evil to even try.I am not against testing/feedback per se; indeed, effective feedback is an essential component of good education. However, I believe that standardized testing is inhumane, ineffective, and destructive. So I don’t recommend trying to improve it, but rather I recommend scrapping it in favour of a more personalized approach.

“But beyond inhumane, it is also extremely ineffective in fostering good teaching and learning.”

Doctors used to say the same thing about medicine. People are too individualistic, disease progression varies widely, and different medications work different ways on different people. There’s no point making some sort of standardized test, because there no sort of “standard person”.

Then John Snow showed them all how wrong headed this can be. As he noted -saving whole generations in the process- for any group of people above a certain number they absolutely do act as a standard. This realization, and another 160 years of backtesting, has turned into a modern science of almost miraculous power.

So do you think that standardized testing in medicine is “inhumane”?

” If you want the system to produce graduates that are operating on a high level in some reasonable ways, then there is lots of measured evidence that we have a big problem”

Lots of *measured evidence* eh? How do you know *those* tests aren’t the ones that are wrong?

With all due respect to all of the teachers who complain about this, if there is a problem with the test then you fix it. Let me give you a commonly used example. The Israeli Air Force uses standardized testing to find pilots. This raises the possibility that the test is wrong, something they *really* can’t afford. So what they do is randomly pass several candidates every year who failed the test. Those who make it through the program are then intensely studied in order to modify the test to make sure that person would pass the next time.

If, instead, they scrapped the test entirely, all of the good questions would go out with the bad. They’d be starting over with some other set of problems again and again.

I’m not sure what you are referring to when you speak of John Snow. Are you referring to his development of anaesthesia? If so, we are talking apples and oranges here. The idea of testing people to see how much anaesthesia is needed during surgery is a far different problem than the difficulty of helping students to learn in schools. Systems of standardized testing in medicine are not the same (and do not have the same purpose) as the soul-crushing standardized tests used in schools nowadays. So, no, I don’t have anything against standardized testing in medicine, but I stand by my points about standardized testing in schools.

Lots of *measured evidence* eh? How do you know *those* tests aren’t the ones that are wrong?Good point! But look, there is a lot of anecdotal evidence that the standard education system is failing in many instances. Just talk to teachers and listen to their stories, which support the measured evidence. I taught mathematics and science, and I have spoken to teachers across North America, and hear the same types of stories from teachers everywhere I go: Students are graduating from high school without being well-prepared for further study. Many high-school graduates can’t read well, and many can’t think well either. When I was teaching at a university (I stopped about 18 months ago), I saw with my own eyes and heard with my own ears how students were suffering. They had passed many tests in high school, including standardized ones, but many of them were poorly prepared for university mathematics and science. It’s not just standardized testing that bothers me, it’s our terrible system of grading. Students accumulate partial credit, and some students manage to achieve high scores thereby, and are lulled into a false sense that they know something, when their understanding is actually quite weak. The intense push for standardized testing in some areas of North America (it’s being pushed into elementary school in some jurisdications in the U.S., and student scores on standardized tests are being used to bludgeon teachers, which is a horrible way to run an effective education system) is only making this worse, because it creates intense pressure on both teachers and students to narrow their focus to what will appear on a standardized test; that is, what is easily tested, but not what is optimal for preparing students to be excellent thinkers. This is my main complaint about standardized tests, although there are many other things to complain about.

Standardized tests are loved by certain adminstrators because they are relatively cheap, they give the illusion of objectivity, and they produce numbers that can be easily processed (although what the numbers mean is almost certainly not what is relevant). They allow students to be easily ranked, and they are now being used to rank teachers too. However, they are counterproductive pedagogically.

Your point about the Israeli Air Force testing is a good one; I admire their scientific approach to “testing the tests” to improve them. But again, we are talking apples and oranges. The Israeli Air Force test is an *entrance exam* to weed out many applicants to select an elite few for a highly specialized training program. I have no problem with this, and if you say this is the best way to select entrants to medical schools, law schools, etc., then I will grudgingly agree; all of these programs (and others) have many more applicants than they can accept, and if standardized tests work for them (i.e., a large proportion of their selected applicants are suitable), then OK. However, the situation in education in general is far different. We are talking about the growing use of standardized tests as a tool in an education system that seeks to educate *all students*. We are talking about the use of standardized tests in elementary school!

I believe that standardized tests are not a good pedagogical tool for helping students learn, and I would definitely scrap them in favour of a system of verbal feedback. This requires teachers who know the students to give them indivdualized, personal feedback. Consider a sports coach, who goes over to an athlete and stops his practice, demonstrates the correct way to do the thing, and then observes the athlete to see if he or she is doing it right now. This is feedback, and it can be highly effective. Giving the athlete a standardized test, or a grade, is silly. What I am saying is that a student learning mathematics is much like an athlete learning a sport; prompt, personal feedback in a loving environment is very helpful, but standardized tests are not helpful in learning.

It would be interesting to transfer some of the same methods (such as standardized testing) that are being advocated by rich businessmen who know nothing about education and have never taught in a classroom (such as Bill Gates, for example) into the workplace. I’m not sure what you do for a living, but imagine that standardized testing were brought into your workplace. Do you think it would be helpful? What changes would occur in the morale of the workers, in the relationships between workers and supervisors? If it’s advisable for school, perhaps it would be good for the world of commerce?

What do you think?

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