# When the tests are powerful and wrong

**A great math teacher deals with dicey math, because it is on the test**

*“I’m providing Test Prep for students who have to confront this type of language regardless of its mathematical validity!”*

One of the trickiest topics in K-12 mathematics is probability. It’s tricky for a number of reasons:

- It’s not a traditional topic. There are not decades of practices to use, improve, or rail against.
- Probability relies heavily on fractions, the gateway between arithmetic and algebra, the single aspect of grade school mathematics that we screw up (in this country) the most
- The subset of probability we teach, simple combinatorial probability, is not a standard part of college probability courses – we don’t have the usual crowd of post-secondary math people poking around and complaining about mistakes.
- Most people teaching probability learned their probability k-12, or from math ed classes that were based in k-12 curriculum. In other words, what we do poorly, we pass down not only to students, but to the next generation of teachers.

So this twitter exchange is with a great math teacher, retired. As department chair he took some of the most advanced classes (everyone does that) and some of the least advanced classes, full of kids who struggle (no one does that). Currently he is doing SAT prep. He prepares SAT-type questions, and I think he solicits the occasional comment.

Great Teacher: A pt is chosen randomly inside larger of 2 concentric circles. If the prob the pt is outside smaller circle is 84%, larger rad:smaller rad=?

JD: Is “random” smooshy here? If I randomly choose an 0 < angle < 360 and a radius 0 < radius < BIG Radius, is that not random?

3rd Party: Good point – ‘random’ doesn’t necessarily mean ‘uniformly random’

GT: Excellent pt re random. The language I used is common on stand’zed tests where “uniform” is implied. Pls reword it in <=140 char!

JD: Why use “random”? Why make it a probability q? 2 conc circ, Large and small. 84% of area of L is not in s. Radius(L) / Radius(s) ?

GT: Agreed but I’m providing Test Prep for students who have to confront this type of language regardless of its mathematical validity!

Probability is really hard. I mean, it’s hard to learn, it’s hard to teach, and it’s hard philosophically. I think the question in question (?) would have been fine with the word “uniformly” added, but actually explaining what “uniformly at random” from an infinite probability space means is very tricky.

Agree completely with “probability is hard” – made worse because it looks easy, and if you are taught it poorly, what you know may very well be easy (and wrong), and many of us were taught it poorly.

For the question in question, I would stick with my recommendation to ask for the ratio of the radii, and leave probability out of it.

Jonathan,

I appreciate your kind words and I share your concerns. Sadly there are far more egregious examples of lack of quality control on standardized tests than this one. The SAT is far more reliable and is of higher quality in general than state, local, publisher and teacher-made tests.

Geometric probability using ratios of areas has been tested for some time now and the term “uniform” has never appeared on any test I’ve ever seen. Your point however is well-taken but I personally am more concerned about overall quality of assessments.

One SAT question from many years ago asked students which proportion among the choices was not EQUIVALENT to a given one. The term “equivalent proportions” has never to my knowledge appeared in a text or on any other standardized test before or after. This one got past the editors and that’s the College Board which is better than most!

My even bigger issue these days is the lack of interest in pedagogy re effective strategies for teaching specific concepts/skills in math. Go search this topic in Social Media and let me know what you find!

Dave

Agree that many test makers are far worse than College Board. You have seen what’s been written about the state tests in New York?

But probability has its own special set of problems, since those who’ve designed high school curricula or tests often have incomplete or erroneous knowledge of the topic.

Leaving the testing aspect aside, here is a salvage for the problem to turn it into a project like the real world math course Tim Gowers has blogged about:

Same set up with two concentric circles. Now, have a darts player throwing darts at the picture. You notice that 84% of the time that the darts land inside the large circle, they are also inside the smaller circle. What is the relationship between the radii of the two circles?

If I’m throwing, then a uniform distribution is probably fair and we get the “textbook” answer. If someone with skill is throwing, where are they aiming? If they are trying to get inside the smaller circle, then higher skill (how would you define that?) implies a smaller inner circle. If they are trying to get between the circles, then higher skill implies a larger inner circle. What if they are trying to hit a target (point, region?) outside the two circles?

If you dare, then extend the problem to discuss weaponry (e.g,. precision guided munitions)? That would introduce considerations around the cost of hitting the wrong target, the cost of precisely locating the intended target, etc.

What do you think?

Sounds like fun for me, but not sure how to do it with a class.

Even for me or you, where’s the data coming from to help us formulate a conjecture about the distribution. Maybe we could get a good player to throw for a while at the bulls eye, and measure the distance from the target?

But if they are good enough, the clustering could be too tight. Go for someone with middling (pardon the pun, haha) skill? Or extend the distance to the target?

There must be a study out there, but I’m game to try without googling

Not quite hands-on but a MonteCarlo-type simulation on the TI-84 for example got my students excited several yrs ago. Yes it’s a pseudorandom generator but students get to see 100 or more pts light up and see the proportion which land inside the smaller circle. I wrote the program but I left it in the margins of my notebook! I gave the program to my students so someone may have it but it’s very easy to code. I’m sure there’s a Wolfram demonstration already out there.

Well it may be fun to watch the results of the simulation pop on the screen there is of course a problem. The code includes assumptions about the distribution which kind of defeats the purpose of doing any kind of simulation.

the only way to write clear problems… in or out of probability… is to establish clear contexts. most “weak” students… and almost all administrators and politicians… prefer *not* have their work clearly understood. it appears certain that “find the ratio of certain areas” has fallen into such bad odor that one can *only* pose such problems (in certain contexts) as a “probability” (with or without the explicit assumption that probabilities can be computed as ratios of areas in such contexts). *you* do the math.

I agree that traditional ratio of area problems now need to be put into more of an applied or real-world context and “geometric probability” is one such vehicle. Questions like the one I posed have been around for quite a while. It may not be Buffon’s Needle but it is one that is covered in many geometry texts. Using ratio of areas to model such probabilities is well established and necessary assumptions are usually implicit. For much better examples go to

http://www.cut-the-knot.org/Probability/GeometricProbabilities.shtml