# GL(s,R)

## May 22, 2010

### SBG: My first step

This fall, I get to teach Finite Mathematics and Precalculus. While I plan on using Standard Based Grading for both, my focus so far has been on the former. A little background on the course is probably helpful.

## Finite Mathematics

At Suffolk, every student has to take mathematics as far as calculus for management and social sciences, calculus for scientists or finite mathematics. On average, I would guess the calculus for scientists class gets two classes of 12 students per semester. The other two, meanwhile, get the remainder of the student body. Consequently, both of those courses expect to run several dozen classes per year (the classes are capped at 25 students!) To avoid anarchy, these courses are coordinated courses. That is, there is a course coordinator who sets general policy, chooses the course content and writes the final exam taken by all students in the course. The course coordinator will come back in a later episode of this saga, so keep him in mind. The content of finite mathematics is all over the place. There are three main topics in the course, separated by midterm exams. The topics are linear equations and linear programming, financial mathematics, and combinatorics and probability.

## The Standards

Before the spring term ended, I got a little excited reading about SBG and read through the entire finite mathematics text, taking notes and writing down standards. I then created this spreadsheet. Although it is about 3.5 months from being finalized, I’d love some feedback on it. My next step was to read through the book again with a copy of the final exam from last year (along with the formula sheet the students get). I noticed that the final exam reduces nearly the entire section on financial mathematics to inputting data into the formulas on the formula sheet.

Do you really want your kids to be the kind of people that won’t attempt to make a pizza from scratch for lack of the 1 tsp of anise seed that has little to do with the overall success of the dish? – Cornally

Problem 1: How do I motivate myself (much less, the students) to honestly cover that section knowing how it is tested?

Next, I noticed that the book is in the wrong order when it comes to the combinatorics and probability: they teach combinatorics first and then probability. The problem is that the interesting questions in the course that use combinatorics are all in the probability section. Wouldn’t it make more sense to teach the probability section first and when you get to a problem that requires $_nC_r$ to simply state that we have a problem we don’t know how to solve, i.e., create a crisis? Now, if the original problem is worth solving, the students have an interest in solving my otherwise contrived problem of choosing n things from r things (although it did seem less contrived before the university made me stop using my Kama Sutra example).

Problem 2: Convince the students to not worry about textbook order. Meh, most of them never open the textbook anyway.

The next thing I noticed is that the book treats the formulas for counting and computing probabilities as distinct formulas, when they are really the same thing. Reading Gerd Gigerenzer’s Calculated Risks made this quirk even more annoying. If you followed the recent New York Times series of Steven Strogatz, then you may already know about the book. If not, check out the article Chances Are from that series. It is stunning how many ways the students (or doctors or lawyers) can avoid mistakes in computing conditional probability by simply computing in terms of frequencies instead of probabilities.

Problem 3: Convince the students to not worry about the formulas in the textbook.

Finally (for now), I was discussing with one of my students how to plot the feasible region in a linear programming problem.

In the diagram above, we plot the following system of linear inequalities:

$\displaystyle \begin{array}{rcl} y & \geq & 2x+3 \\ y & \geq & -3x+4 \\ y & \leq & 6 \end{array}$

The feasible region is where all the shaded regions overlap. Using the picture, it isn’t so hard to spot the region. However, imagine drawing this with a single pencil. All of a sudden, it isn’t as clear. The book suggests that instead of shading first, to use arrows that point in the correct direction and only after finding the region where all the arrows simultaneously point do you then shade the feasible region. After explaining the process to my student, he informed me that in Vietnam (his country of origin) the regions are sketched in reverse.

Essentially, you shade the regions you don’t want and are left with a hole: the feasible region. You only need one color and you can avoid drawing the arrows.

Problem 4: Convince the students that they won’t be marked down on the final exam for drawing their solutions differently than the every section of the course.

That is enough for now. The next post will be my first Tricks of the Trade post and then I’ll come back and talk about my learning a foreign language fetish and how it relates to teaching (I doubt anyone can correctly guess where I’m going with this!) Sorry for the length; there is a lot of thought bottled up. The next post will definitely be shorter.

1. Hey there 🙂 I’m a middle school math teacher, and I just switched to SBG this year. It’s made a huge difference in what my grades look like and what they represent — now I truly feel that their grades represent their understanding of each concept. Remediation has become a lot easier too, though I need to channel Shawn Cornally more and say, “I am not your grade’s babysitter.” Anyhow, good luck — I think you’ll like it a lot once you’ve got your concept list (tentatively) nailed down for the semester 🙂

PS. I really like the idea of shading the non-desired side of inequalities, though I think my kids would be confused on standardized tests…

Comment by Meg Claypool — May 23, 2010 @ 12:10 am

• Thanks for the great feedback and encouragement, Ms. Claypool (and for being the first ever comment on my blog!) Have there been any issues at your school with your students telling other kids about your system and those kids complaining to their teachers that they don’t get to retake quizzes as many times as they need to?

@inequality shading: When I first mentioned the idea to one of the other teachers, I was told not to teach it that way as the students would be unable to get help from the book or our mathematics support center. I see the point (which is related to your point about standardized tests) but something doesn’t sit well with me. How can I intentionally make things harder (at least from my perspective) for my students and feel good about it?

Comment by Adam Glesser — May 23, 2010 @ 5:49 am

2. Since both of the 8th grade math teachers (myself and my colleague)in my school are using the same grading system, there haven’t been any complaints 🙂 Right now, although I’m firmly committed to SBG, I’m struggling with finding the perfect implementation. I originally got the idea from Dan Meyer (http://blog.mrmeyer.com/) and only later discovered Shawn Cornally et al.

Dan’s twist on the whole thing is that students’ scores never go down — if they ever understand a concept, then having a bad day on the next quiz won’t impact their grade. This is very appealing when presenting this idea to the students. However, I’m having a huge problem with cheating. (I’ve only caught students in the act a couple of times, but the circumstantial evidence has become pretty damning.) Of course, the cheaters can’t replicate their earlier scores once I move them away from whoever they’d been cheating from. But using my current system, that’s ok (from their point of view), because they’ve already “earned” a fairly high score. So I may have to make a change to a more Shawnesque SBG. In fact, I think I’ll wander over to Dan’s blog and pose to him the question of how he handles that situation 🙂

@inequality shading: The problem is that mathematics has loads of conventions, and setting my students up with a method that is contrary to a current convention is setting them up for difficulties. Are some students smart enough to handle using one method for themselves and recognizing another method on the state tests? Of course, but lots more of them would be confused 😦 Too bad, because it seems like a far superior method 😦

Comment by Meg Claypool — May 23, 2010 @ 4:39 pm

• Philosophically, I’m inclined towards the ‘Shawnesque SBG’ version where grades reflect current understanding. Having said that, I would love to hear Dan’s response to your question as he probably has some masterful way of short-circuiting the problem.

I’m sorry to hear about the cheating issue. In those cases where the (circumstantial) evidence is so damning, would it be practical to express those concerns to the students and let them retest? In college, it is much easier: if I can demonstrate cheating, the students automatically flunk. I imagine your situation is a bit more delicate.

@inequality shading: Your point is well-taken. While I have serious issues with those tests, you certainly don’t want to set the students up to fail on them.

All the best,

P.S. I have a four year old son and he very much appreciates the Greek alphabet sheet on your website. Thanks for putting up such a nice version.

Comment by Adam Glesser — May 24, 2010 @ 6:13 am

• Well, I emailed Dan, and his response is that he’s pretty sure his kids aren’t cheating. He has never taken points away because of cheating.

My evidence is almost always circumstantial. Since I have kids seated 3 or 4 to a table, I’ve been told (by more honest students) that they slide their papers under the privacy shields to each other (which is really hard for me to catch them at). I’ve only actually caught kids once and made them retake, but other than that it’s all “I really think you cheated by I don’t have any solid evidence…”

Next year I’m moving to another classroom, with individual desks, rather than shared tables. I’m sad to give up the tables, but I don’t think my 8th grade students are morally mature enough to handle such close quarters without cheating. Like the saying goes, “I can resist anything except temptation.”

@ Greek Alphabet: extra points if you can sing it with him to the tune of the alphabet song 🙂

Comment by Meg Claypool — May 25, 2010 @ 7:17 pm

Welcome to the crew! I’ve been working/evangelizing SBG for the last couple of years now and am excited to hear about your journey. I’m really excited to see how it’ll transfer to higher ed. Keep at it and if you’re having probs we’re a supportive community. You can find that whole group of us you listed in your first post on twitter as well. Good luck.

Comment by Jason Buell — May 23, 2010 @ 10:09 pm

• Hi Jason,

Thanks for the hearty welcome! I look forward to finding the larger support community because I will certainly need it. Also, thanks for becoming (to my knowledge) my first Reader subscriber; even my mother hasn’t got around to doing that yet! I hope I can keep everyone interested.

Cheers,

Comment by Adam Glesser — May 24, 2010 @ 6:18 am

• According to my Google Reader there are four of us. Just keep commenting and asking questions. If you’re on twitter I’m @jybuell. Shawn, Sweeney, Sam, Kate, David, Dan, are all on there too.

PS – My mom is addicted to Farmville so I can’t get her to read my blog either.

Comment by Jason Buell — May 24, 2010 @ 11:01 am