I want my pre-calc students to make as much mathematics as they can. When they solve a problem, I want their excitement to carry them into the next problem.

I also want them to practice getting their ideas down on paper, with correct spelling and grammar. They are seniors, and their great ideas won’t get them very far if they can’t express them cogently. Their professors and employers will take incomplete, ungrammatical, mis-spelled, mis-punctuated, mis-capitalized sentences as evidence of ignorance.

The rules in class are that grades will be based only on finished papers. Finished means (among other things) a sensible problem statement, explanation and conclusion, with no grammar or spelling errors. (I secretly allow one error per paper.) I give lots of feedback on every draft, but no credit for unfinished work. Result: as the term ends, the relaxed pursuit of new problems has given way to a grumbling, resentful process of revision after revision. “We already have to do this in English and History. We understand having to write it up once, but why do we have to make perfect drafts? This is math class, not writing class!” They’re stressed: students with ten papers in progress have no credit yet. And I sympathize. I remember this from grad school. The fun part was figuring out the solution and a neat way to explain it. The long, boring, maddening part was the subsequent rounds of reviews, revisons revisions, rewriting.

So why am I doing this? I guess because my experience is that everything I start teaches me something, but only the things I finish move my life forward.

Is that the most important lesson at this stage of their lives? If it is, am I teaching it the right way? Turning a fun class into a tedious chore seems like a bad idea. But letting them skip off to college in the habit of turning in papers that would embarrass many 7th graders seems like a bad idea too.

Maybe I should ask them.

In the meantime, if you find any errors in this post, please let me know.

Decades ago, when the Finnish school system was badly in need of reform, the goal of the program that Finland instituted, resulting in so much success today, was never excellence. It was equity. -Anu Partanen

Is it a contradiction that I simultaneously feel so positive about this article, and so grateful for my wonderful, private, education? Maybe not. Maybe I just think everyone should have what I had, and that there’s no good reason everyone can’t.

(Thanks, Anna!)

To my students—most students?—mathematics is “Substitute, then simplify.” It’s all math has ever been. I have been aware of this, but my response has been inadequate.

• Item: I would put two numerical examples on the board, then a third using variables, crowing that “if you do the same thing with only letters, then you’re doing it for all possible numbers at once! Do it once or twice with numbers, then do the same thing with variables! This is called generalization!” Result: I had one student do a numerical example similar to the problem above, then go on to say “I wanted to solve it with variables, so I made 16=U and 49=V, so the answer is .”
• Item: The idea of a derivation is entirely absent. Later in the year I juxtaposed three student examples of finding a distance to show them that the distance formula is simply the Pythagorean theorem. Students were flabbergasted. I got compliments on my lecture. They had never seen anything like it.
• Item: After developing a meaning for the distance between a point and a line, I asked them, “What is the distance between and y=d?” and gave them some time to work on it. They were totally stumped. “Ok, let’s draw a picture.” Blank stares. “What’s up?” “We don’t know what d is—you have to know what number to use!” Ok. Let’s graph y=2. Fine. How ’bout y=-4. Great. What should we do for y=d? Someone, quiet, tentative: “Draw a flat line?” (indicating horizontal with his hand). Does it matter where I put it? No. What about here on the x-axis? No, don’t do that, that makes it look like zero, and it might not be zero.

In my mind, once you do something with specific numbers, it’s just one more small step to do it again with symbols. After all, it’s the same thing. But that’s the rub: to them, it’s not the same thing!. Torigoe (Thanks, Mylène) has got me thinking about what might be going on.

It never occurred to me that carrying symbols through a problem is more demanding than substituting numbers into as many variables as possible, as soon as possible. By stopping up all those leaky variables with numbers, I can stem the flood of confusion behind a nice high dam of something with just one variable to solve for. The structural relationships among whatever I’m modeling (say, distance between objects on a plane) are eliminated, so I don’t have to think about them anymore.

Anymore? At all! This explains everything! It’s why my precalculus students can’t articulate relationships between abstract quantities. They’ve never had to! So they don’t know how.

At least not yet. The technique of “give one specific problem, then a second, then a general one” and hope they make the leap has not worked. So how can I build the capacity to reason with symbols?

We drew a circle and measured the diameter and radius. The diameter was 3 and radius was 1.5. The formula was C=2ΠR so we multiplied 2 x 3.14 x 1.5 and it equaled 9.42. Then we divided 9.42/3 (diameter) and it equaled 3.14. Which is C/D and was the info that was given to us. So we proved that circumference is C=2ΠR. -A student proof.

The proof I was expecting was something like “Since Π is defined as C/D, then C=ΠD, and since D=2R, then C=2ΠR.” What’s the feedback that will cause them to be concerned about thinking in circles? The best (only) idea I’ve come up with is: “If you had multiplied 2x7x1.5 to get 21, then divided C/D = 21/3 = 7, you would have shown that Π=7.” But I have ’till Jan 4th to find something better. Any suggestions?

Be kind; everyone you meet is fighting a hard battle. -unknown

The good news is, I have 110 papers from 50 students.

The bad news is, I need to read and comment on them all.

Two missiles are approaching one another moving at 9,000 mph and 21,000 mph, respectively. How far apart are they 1 minute before they collide? 9,000/60 gives 150 miles per minute. 21000/60 gives 350 miles per minute. Put those together and you get 500 miles in two minutes. Because we want to get one minute, divide 500 by 2. The missiles are 250 miles apart before they collide. -A presentation by two students.

The class buys it. I express confusion: Isn’t there just one minute we’re talking about? Exactly! The whole class responds—they’re so pleased I finally get it—That’s why we have to divide by 2!

By the norms of the class, if the community buys the argument, the presenters can submit the paper. It still has to pass review, however, and I’m the reviewer.

The goal, of course, is to avoid the completely useless path of simply telling them the answer is 500 and they’re just wrong. As far as I can tell, it is precisely as obvious to them that the answer is 250 miles as it is to me that the answer is 500 miles. Their answer makes sense to them, and my answer doesn’t. That’s what I need to reverse.

So I offer my burden up to the blogosphere: What question can I ask my two junior colleagues that will help them see what I’m talking about? Or, as a distant second choice: what’s a clear way I can explain to them why it’s 500 miles and not 250?

Schools in the suburbs are different. I went to a football game in the suburbs, and they had a band. The players came out and then this marching band came on. A band! Why don’t we have a band? -One of my students

I told her not to lose sight of that question as she went through college. I told her it was the most important question in America right now.

Did I overstate?

I just realized that when I put a parent or guardian’s contact info into my phone, if I put the relationship in the “Title” field and the student’s name in the “Company” field, two great things happen.

1. Those fields are included in the search function, so if I can’t remember Jimmy’s mom’s name, I can search my contacts for Jimmy Smith and get a list that includes
   • Jimmy Smith
   • Bernice Simpson (mom, Jimmy Smith)
   • Harold Edgerton (step-dad, Jimmy Smith)
2. Then later when Bernice Simpson happens to call, if I still have not properly linked her in my mind with Jimmy Smith, my caller id helps me out by flashing that it is “Bernice Simpson (mom, Jimmy Smith)” calling.

If you are picking software for SBG from scratch I recommend ActiveGrade. I’m very happy with JupiterGrades but ActiveGrade is designed specifically for SBG and has a lot more flexibility for SBG. I’m using JupiterGrades because (a) it’s a very good program in my experience and (b) my school is using it.

First some notes:

1. JupiterGrades does offer a Standards Based Grading setting for classes, which you can read about in their help files. I was planning to try that this year, but it must be set up by an administrator. This is probably a good thing: you want teachers and administrators all to agree on what the standards are going to be for a given course. But in my case I’m the only one teaching my courses and my administrator is busy. She tried to set the program to let me do it, by giving me “Y” permission, whatever that is, but when I tried again, the program said I also needed “S0″ permission, so I gave up. Instead I’m just using JupiterGrades normal grade setup screen to get what I want. For better or worse, if you teach with JupiterGrades, the software will let you do what’s described below without any centralized coordination about standards.
2. I wrote about this last year. If you’re wondering what SBG is, that post has some links to great writers on the subject, so if you’re new to SBG, start there. Then come back to this post to get into details about setup screens and so forth.

Ok here goes:

1. Set up the grading options.

On the Setup>Grading Options screen, do this (click any image for a larger version):

Comments:

• Everything on the left is pretty normal.
• Under “Special Marks”, you translate what your “rubric grades” mean in terms of percentages. I’ve stopped giving zeros, so my map is 4=100%, 3=80%, 2=60%, 1=50%, and 0 (blank)=40%. So if I give a “3″ on a standard, that student has an 80 for that standard. But I’m always messing around with that. In any case, this is where you set those.
• I don’t use categories because standards are the only thing I grade. If you want X% to be standards, Y% to be something else, you’d use categories to set that up.
• You’ll see I have the weightings as 80% of the total grade is based on 1st quarter and 20% on the exam. Here’s why. Students final grades are actually based 80% on whatever their average is across all standards at the end of 4th term and 20% on their final & portfolios. But if I put that in now, then their grades during first term won’t show up. So when each new term starts, this changes. See “New Term”, below.

2. Put your standards in the “Assign” tab.

Here’s an example:

Notes:

• Use “Input as: Grades” instead of Percent so the program will translate your 3′s and 4′s into 80%’s and 100%’s (or whatever you chose on the grading options setup screen above). If you use “Percent”, then when you give a cherub a 4, it will go in as 4%.
• I didn’t show it in the screenshot, but in the Description box, you can put a few sentences describing the standard. Then students can see the description as well as the title when they look at your class in JupiterGrades.
• Last year I put in two lines for each standard, and kept students’ best-ever scores as well as their most-recent scores. I’m not doing that this year: too many students had their best score in November and awful scores in May and ended up with a grade that was the average. I don’t think those students’ grades reflected their level of proficiency. So this year it’s most-recent only (with lots of chances to improve if your most-recent was low).

Using it day to day.

Imagine I give a quiz on 9/27. On that quiz I might have 2-3 questions, one of which assesses the standard “Explain and use the product rule.” In the screenshot above I have entered grades for each student. I use the comment field to say when the standard was last assessed and how. So the screenshot also shows that a few students got reassessed after 9/27, by showing their proficiency (or lack) on that standard on a homework presentation or a re-assessment.

How the grade is computed.

When you click on a particular student’s name, you see the list of all standards entered so far, and the student’s grade on each one. The overall grade at the bottom is just the average of all the standards grades (after they’ve been translated to percents). So this student has been assessed on 2 standards, receiving a 4 (100%) and a 3 (80%). So far her grade is a 90%, an A-:

When a new term starts.

Each term students pick up right where they left off. If in first term we covered 6 standards, then they each have 6 grades (one for each standard). When second term starts, I want them to start out with those same 6 grades on standards 1-6. Over the course of second term, they’ll have the opportunity to improve their grades on those six, as well as working on new standards.

To do this takes 3 steps:

1. Start a new gradebook for the new term.
2. Use the Setup>Import/Export feature of JupiterGrades to export the term 1 assignments and grades from the term 1 gradebook and import them into the term 2 gradebook.
3. On the Setup>Grading Options page, switch the Grading Period Weight so it’s 80% 2nd term, 20% final.

Repeat for third and fourth terms. By the end of fourth term, students will have their most-up-to-date grades on every standard in your fourth term gradebook. Their grade for the year will be based 80% on their end-of-year standings on all standards, and 20% on their final and portfolio.

At least that’s how I do it. Hope this is helpful.