In which I confess my secret

(This entry is about one of my major failings as a teacher. I assure you that I am not fishing for compliments or anything from the community here--just needing to get it off my chest.)

It's finals week at our school and this brings up one of the most shameful aspects of my classroom.

My students consistently do poorly on standardized tests.

The two sections of precalculus that I teach have, in local lingo, Common Course Exams. The district writes the exams for all precalculus students in the county to take and this counts as the final exam for every precalculus course (no curve). It consists of 50 multiple choice questions and we teachers are not meant to look at it until we are handing it out to students to take.

This year, my averages were 41% (high of 74%; 21 students took the exam) and 45% (high of 70%; 19 students took the exam) for the two sections I teach.

The issue doesn't just start there, though. For other courses I've taught with these CCEs in the past, the same issue has arisen. And most ashamedly, in my students' AP scores. Last year, for example, not a single one of my 19 Calculus AB kids passed with a 3 or better on the AP exam.

What makes me feel worst about this is that I don't know why this is an issue for my classes.

I have a list of excuses below, but ultimately I feel like I am: 1. Misrepresented by the data (which has cost me the chance to teach calculus next year since the principal is not happy with the scores from my previous classes) and B. Letting down the students who do seem to be trying.

Some excuses:

1. I do not get to see the exam before it is given, so I cannot easily prepare students for what is expected
2. I teach mostly seniors, so many of them are not focused at the end of the year
3. At our school, seniors can be exempt from the final exam if they: A. Have an A in the class and B. Have not missed more than 6 days of school. This will bring down my class averages since the best students will not be taking the exam.
4. Alternatively, students may be exempt from the final exam if they take the AP exam (regardless of how seriously or whether they are seniors or what their grade is in the class or attendance). This artificially lowers my AP scores since many students (who admit it before going in) will sign up to take the AP exam just so they can be exempt from the class exam.
5. Many students who are already failing in my class have just "given up" on the class and do not even attempt the exam. Still others play the numbers game and calculate things like, "I can still pass the class (or maintain a B) if I get a 25% on the exam."
6. The exam was 50 multiple choice (4 choices) questions to be finished in 2 hours. Every question missed lowers the grade by 2%, so to get an A, for example, you can only miss 4 questions.
7. The same CCE is given to both honors and standard levels. What's more, the exam was written by 3 teachers who have exclusively honors level classes. I teach all of the standard level classes at our school. The honors teacher had an average of 66% (high of 86%)

Rebuttals to excuses above:

1. This is an issue for every teacher and not all are having the same results as myself.
2. This, too, is not specific to my classes. Also, if the students know the material well enough, this shouldn't be a very big issue
3. In the class with the 45% average, I had at least 4 juniors with an A in the class who were not exempt.
4. This is a big issue and a problem I have with our district's policy. It still does not account for all students and having none pass last year is abysmal.
5. This does not account for all students, and I know many of them worked hard to review and of the 2 hour exam, most took at least 80 minutes, so they thought about it some.
6. Most students took between 80 and 110 minutes to finish the exam. Nobody was still working at the bell, so time did not seem like a true issue (although they may have been watching the clock and gave up a few minutes early).
7. We are still in the middle of exams, so I don't know what other numbers from the county may be like. The honors teacher's numbers seem more reasonably explained by these excuses than my own.

So, I don't know what to do about this. Whenever I get results like these back (which is too often), I get depressed and want to give it up. Not teaching, because I couldn't give that up, but maybe these "new fangled methods." If I'm going to be judged on standardized tests, maybe I should go with the tried-and-true methods of drilling homework for grades and a standard quiz/test system that gives students one high-stakes chance to get it right.

I don't know. I'll have all summer to get over it (until AP scores come in, I guess) and should come back around. I know my students know the material. They have it mastered from what I can tell in class, but they can not show it when the data counts. I took it as flukes the first couple years of this happening, but now it has become a pattern.

Altruistically, I know my students are learning and enjoying learning. They master all of the skills on the curriculum and then some. They are challenged and enjoy me as a teacher. An interview with ANY of my students (even those who gave up on the class and failed) would let you know what kind of teacher I am. Ultimately, they learn and that's what counts. I just hate that I may end up a martyr for it.

Baby's First WCYDWT

Our little baby's all grows up!

A couple weeks ago, it was (unexpectedly) senior skip day at my school. I found myself with no more than a half dozen kids in class and was torn about whether to actually teach/review something or do something more “fun.” (It’s sad that I, too, separate the two too often.)

This was my opportunity to try out something I’ve seen all over blogs and have wanted to do, but never got around to it. It’s a classic story in math circles and that’s all I told on that day (not having more ready), but I thought I’d update it with more modern visuals. You will already recognize the story by having glanced at the pictures below, but many students have not heard this yet by the time I see them in their lives, so it will still be novel to them, but the questions may venture in different directions.

I think a “pure WCYDWT” would mean just presenting the pictures. Of course, this one comes with a cute and somewhat compelling story (especially if you include the “ending” that the emperor of India realized what was going on by about the 20th square and executed the man--these age kids love death), so you should gauge your own class as to whether you tell the story or not. I think the lesson will go similarly either way.

If you’re not tied to the story, though, doing this with pennies would be another interesting way to go (especially if you don’t mind getting a little political and talking about our national debt and other large money numbers). There is much less variability in the measurements and the visuals could be created more easily (and is more familiar to more students than rice). There are more extensions to other forms of currency (with pictures to be found on the internet fairly easily) and they all stack fairly nicely.

Here is the opening picture:

And some other angles:

Some other useful pictures that can be revealed if questions take such turns:

A few grains for scale

And to get the other side?

Different kinds of rice

40kg bags of rice

Somewhat political in nature, but good visualizations of rice in piles

Rice plant

Rice paddy

I did it in my two standard precalculus classes and they each went in slightly different directions with it. In one class we got to the large scale by using the weight of a grain of rice (see the bottom of this page), then envisioning a 10lb bag of rice as a backpack. In the other class they didn’t know what a 10lb bag looked like (and google images is blocked at school), so we happened to find that the average length of a grain of rice was about 7mm and then just assumed it was a sphere (clearly too big, but we justified it by saying it puffs up when cooked and that might account for some of it and we truncated most of our decimal answers).

After we got to the large scale, the class who used the weight was more interested in the ability to produce the rice. The second class was more interested in how to consume the rice.

Our district is VERY strict about internet censorship, so my computer is the only one in the room that can access the internet (the kids could use their phones, but that’s also against school policy). Even my computer is blocked from such useful resources as google images and wikipedia, so we had to make do with what we had. So, I looked up what I could on my computer, but this would’ve been much more fun and interesting in smaller groups of students or if each student had his own computer to pursue his own questions.

By the nature of the problem I couldn’t get a picture of “the answer” so that’s perhaps somewhat unsatisfying. My students trusted the basics of the math, though, and I didn’t have any problems with this in class.

The questions I expected (and they asked)::

1. How much rice is already there?
2. How many are on the last square?
3. How much rice would it take to finish the board?
4. How long did it take you to make that?
5. How much room would that take up?
6. What’s the volume of a grain of rice?
7. How big is this room? (counted 1’x1’ floor tiles and cinder blocks on walls)
8. How big is the school? (some tennis athletes in the room said it was 0.7mi to run around, then we estimated school has a square floor plan)
9. How big is Tennessee?

1. Along the line of thinking, “Could we grow that much rice on the planet?”
   
   1. How does rice grow?
      
      1. How many grains does a single plant produce?
      2. How much room does a plant need?
      3. They grow in water? Can we look up a picture?
      4. How much water is in a “paddy?”
      5. Where can rice grow?
   2. How much rice does China already grow? How much in the US?
2. Along the line of thinking, “What would the guy do with it?”
   
   1. How much rice do people eat per year on the planet already?
   2. How do we already store rice?
   3. How big is this guy’s chess board?
   4. How long would it take them to count them all out?
   5. How much bigger is a cooked grain than a regular one? (my wife found this article)
3. Along the line of thinking, “Let’s go on a tangent trip!”
   
   1. Do birds really explode if they eat uncooked rice?
   2. If I make some rice, can we have a party? Can we order Chinese food?!

Some of my own questions brought up by theirs:

1. Does rice dumped in a pile always form a similar cone? If so, what are the relative dimensions? (thinking about calculus related rates problems here where that is often a given in the setup)
2. I know there are experiments that do things like “rice avalanches” to simulate landslides and such. I should look into this for fun tangent ideas to this.
3. If I wanted to get the exact volume of a rice “ellipsoid,” how could I derive that formula? (And would it be accessible to precalculus students)
4. Obvious packing problems.

Other interesting things:

• When I asked for estimations on how much rice there would be in total, all students thought it would fit within our school and most thought it would fit in our classroom. So, the end result was both interesting and shocking.
• Many students were bothered by the “estimation” parts of this and wanted a single, correct answer as in their entire math career to this point.
• While looking up the length of a grain of rice one of the first hits was for a math “word problem” that said something like, “If the length of a grain of rice is 1mm....” I told the class, “Well, this is coming from a word problem, so I’m not sure if we can trust this. Let’s keep looking.”
• Most students wish we could do this every day. It was interesting and they learned! The sample size for my trial, though, was limited to those students who actually come to school on senior skip day, so this lesson may go differently in another classroom.
• At some point in the year I like to talk about what you can do with a math degree and what math research looks like. I could tie the Knight’s Tour in to this problem.
• My other math-chess related story is about how Alan Turing wrote a computer program to play.
• I did my undergraduate studies at Rice University, so this was particularly fun for me!