[**BTW**: I updated the SBG prompts below with some answers from the comments.]

In addition to the material I facilitated on instructional design, the staff at Colchester High School wanted to work on their implementation of standards-based grading. Happily, they had already agreed on the fundamentals:

- We should assess students on what they know now, as opposed to what they knew when we first assessed them.
- Assessment should be atomized to the point that it empowers teachers and students in their remediation.

This left me all the creative, interesting parts. We talked about reporting methods for keeping students apprised of their progress, both individually and as a class. We talked about the effect of SBG on retention. Then we picked a concept and had pairs come up with a score of 1, 2, and 3.

We debated productively about marginal scores — when a 2 turns into a 3, specifically — and concluded that, in a system this forgiving, we’d rather underestimate a student (who could return to improve her score whenever, wherever) than overestimate her.

We discussed, afterwards, how to construct valid, manageable assessments. I gave them four test questions, each of which, in its own way, invalidated what it claimed to measure or was unmanageable at scale. I’ll leave them here. Feel free to kick them around in the comments.

Alex:

The trouble with the two-step equation problem is that it’s also an intimidating decimal arithmetic question. If a student fails it, you don’t know which skill needs work.

The issue with the Law of Sines / Cosines problem is that you do not have to use the Law of Sines / Cosines to solve it. A student can get those right WITHOUT using the Law of Sines / Cosines, especially the 30-60-90.

Also, the concept is too broad. If a student has a 2/4 on “Law of Sines / Cosines,” how do you know which one to remediate?

“Quadrilaterals” is also too broad a concept. If a student has a 3/4 on “Quadrilaterals,” do you know what the student knows about quadrilaterals? Which ones she understands and doesn’t?

We decided “Linear Pairs of Angles” is too small a concept. If every concept were this granular, we’d have several hundred concepts to manage by semester’s end.

## 15 Comments

## Steven Peters

November 3, 2010 - 8:09 amHere’s my guesses for the problems with each question:

I don’t know what the problems are with Solving two-step equations or Linear pairs of angles.

Law of sines / cosines:

The law of sines / cosines is very general and works for all kinds of triangles. The examples here are special 30-60-90 and isosceles triangles. A student may focus on the specialness of these triangles rather than using the law of sines / cosines.

Quadrilaterals:

Again, these are special quadrilaterals with lots of symmetry. Students may focus on the symmetry rather than approaches that work for asymmetric quadrilaterals.

## Jason Dyer

November 3, 2010 - 9:40 amPresumably more students would get the equation wrong due to division error than procedural error. This hits a philosophical argument though, because if students are only exposed to “easy” numbers they start to shy away from the decimals and fractions.

If I was doing full on SBG I would likely grade the decimal error in a separate category from the procedural one.

Linear pairs of angles is another philosophical one. A student could use vertical angles and then subtract from 360 and divide by 2 (rather than the straightforward subtract from 180) but a.) it is unlikely the student knows how to do this but not the simpler way b.) this doesn’t strike me as a terrible lack of understanding and c.) one side in a linear pair can always be extended to make an intersection so this process always works. If for some reason a student was deviating from the expected by doing this, I likely would mediate with conversation rather than points, as I wouldn’t want the student to get the idea the other way — or any particular mathematical way of doing something — was “wrong”, just harder than necessary.

## Chris Sears

November 3, 2010 - 11:36 amWith the pairs of linear angles, I am guessing that the problem is that the problem is asking for only one angle. Getting all three of the unknown angles would show more understanding of the concept

I’m still stumped on the two-step equation.

Steven gave the same answer on the remaining problems that I would.

Is Tom Messner still doing weather on WPTZ (the local NBC station in Colchester)? I’m originally from the other side of the lake. Tom would give his weather reports with too much enthusiasm for 6am and -25 degrees.

## josh g.

November 3, 2010 - 11:52 amMy problem with grading prerequisite-knowledge errors separately from the actual topic is, how many standards do you file for such things? Or do you make a “Prereqs” standard and just stick lost marks in there as a catch-all?

I ask this not because it’s a bad idea, but because I’d love to know a good solution to this problem.

The decimal numbers do make me wonder, though, if it’d be worth taking a lesson or two just to reinforce the concept that “A number is a number is a number”. Give them simple equations to solve, then the exact same equations with decimals, then throw in radicals or something … something like that to reinforce that category so that non-integer examples don’t scare them anymore.

I dunno, I think that needs a more compelling solution too, but I’m stumped right now.

## Curmudgeon

November 3, 2010 - 12:42 pmYes, Tom is still way too cheerful for that early in the morning.

## Alex

November 3, 2010 - 3:25 pmThe trouble with the two-step equation problem is that it’s also an intimidating decimal arithmetic question.

If a student fails it, you don’t know which skill needs work.

## Jon

November 3, 2010 - 4:27 pmAre students allowed to use a calculator on the two step problem? If so, then it’s a matter of procedure and the calculator can be used to ensure the proper arithmetic.

As for the linear pair, students may forget that linear pairs are supplemental. Not usually a big problem.

The quadrilateral problems arise from the sum of interior angles = 360. Parallel lines and angles on the same side of the transversal being supplemental for the isosceles trapezoid. Recognizing an isosceles trapezoid is also an issue. (Silly things as adults see them, but not so for students.) Remembering non-vertex angles are congruent in kites is an issue easy to overlook for students.

For the Law of Sines and Cosines problems the biggest issue is remembering the formulas, especially for the Law of cosines. Students have a hard time determining c, the variable for which the L of C is solved. It’s more perplexing if the vertices are labeled differently.

Don’t know who Tom is, Curmudgeon, but -25 degrees and cheerly should get him thrown into the lake! ;D

## Mike

November 3, 2010 - 5:00 pmThe problem I see with the Quad and Triangle questions stems from the minimal amount of information given in each question…”just enough” to apply the Law of Sines or Cosines. A student who had a mediocre grasp on these questions could stumble onto the right answer and look like a star, especially the trapezoid problem.

Throw some more sides and angles at them to sift through…make the triangles non-right and scalene…

## Tara

November 3, 2010 - 6:16 pmDan, thanks for the time you spent working with our department. I love seeing the view from my classroom on your blog :)

All of the questions you posed made for some great conversations. The best part is that we’re still having them both at CHS and here online!

## Erick

November 4, 2010 - 3:32 pmThe issue with the Law of Sines / Cosines problem is that you do not have to use the Law of Sines / Cosines to solve it. A student can get those right WITHOUT using the Law of Sines / Cosines, especially the 30-60-90. In fact I expect my students to solve 30-60-90’s quickly with minimal computation (if any at all). If these triangles are intended to demonstrate that the student knows the Law of Sines / Cosines, it does not do that.

## Pam

November 5, 2010 - 2:22 amThe quadrilateral problems are surely not intended to represent generic quads. I would expect my students to know that symmetry is the correct/easiest approach to these two problems.

Jason Dyer hit on the way that about (maybe) 10% of my students will solve for linear angles, no matter how often we’ve worked with 180. But, if they use the implied circle’s 360, I’m almost as satisfied, and would never dream of giving less than full credit.

I would not use the Law of Sines/Cosines for either of the problems posted here. Would that mean that I would not receive full credit? Is a specified method intended to be a REQUIRED method?

What is the two-step equation intending to test? And is it the only problem that is testing two-step equation solving? I would need to know those two answers before I could comment on the problem itself.

## Dan Meyer

November 5, 2010 - 9:47 amI added a few answers to the post proper, if anyone’s interested. Thanks for your thoughts.

## Bill Tozzo

November 6, 2010 - 6:13 amHi Dan:

I’m rteading Ken O’Connor’s book, “15 Fixes for Broken Grades . . .” I’m really starting to like what I’m reading. Any other stuff you could recommend?

## Dan Meyer

November 7, 2010 - 5:59 pmI enjoy the folks on my blogroll (to the right there). In grad school, some of my favorite academics are Deborah Loewenberg Ball, Magdalene Lampert, and Jo Boaler. I’d pick up whatever they put down.