A twitter colleague, @druinok, posed the following challenge, which prompted me a great deal of thought:

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How do you convert from standard-based grades (often on a 1-4 or 1-5 system) into a percent score required by a gradebook or school district, while still maintaining a sense of what the 1/2/3/4(/5) system means?

While the formulas involved may be too complicated for a gradebook, I approached this problem mathematically: can I create a map from the SBG 1-4 system to the grades I believe they should represent? My understanding of the SBG system is as follows, which informed my selected points and function choice:

• 4 = exceeds standard; demonstrates complete mastery and conceptual understanding with no nontrivial errors
• 3 = meets standard; demonstrates mastery of a skill/topic, perhaps with minor errors
• 2 = approaching standard; shows some understanding, with major (or many minor) errors
• 1 = below standard; shows little or no understanding
• 0 = did not attempt

In my conversation with @druinok, she stated (and I concur) that a student with all 3s has met all standards and should receive a B+/A-. All 4s should clearly represent 100%. In my opinion, all 2s should be enough to scrape by with barely passing the class (60% or D); some may argue that since that student has not shown mastery of the standards he or she should receive a failing grade.  A student with a 2.5 average (half twos and half threes, e.g.) should receive a C. A student with 1s should not pass. In my district, the lowest grade we can assign on a report card is a 50, so a 1 average yields this.

In playing around with the numbers, they seemed to fit naturally into a symmetric pattern around 2.5 = 75%. This in turn prompted me to find differences and look for a cubic equation with inflection point around (2.5, 75).

• 1 average = 50%
• 2 average = 60%
• 2.5 average = 75%
• 3 average = 90%
• 4 average = 100%

My first attempt is odd since it reverses the independent and dependent variables (x/y). My second attempt is more understandable and develops the equation $-\frac{20}{3} x^3 + 50 x^2 - \frac{280}{3} x +100$. Its graph is shown below.

I also tried a third attempt with different points (3=85% & 2=65%) since I was slightly unhappy that the above function from my second attempt was not monotonically increasing on the interval [1,4]. Still, my second attempt is closer to my understanding of the 4-point scale.

For what it’s worth, I also tried fitting a 5-point scale to percentages.

Anyway, that was a bit of fun. But the formulas aren’t perfect, and are certainly a bit complex for gradebooks (the original challenge).

So my main point in writing this post was to ask: those of you who are standards-based grading aficionados, how do you handle the conversion to a percent or letter score?

Certainly any solution, whether it be a formula like this or a more holistic approach like “a student must meet all standards with a 3 or higher to achieve a grade of A, etc”, must involve clear and open communication about the grading policy with students & families.

Filed under math, teaching

### 14 responses to “Standard-Based Grades Into Percentages?”

1. I saw @druinoks question too and I’ve been playing around with % scales for a while too. While I still think a clear rubric detailing to the students/parents what each number 0-4 or 0-5 means is most important, we unfortunately have to still report grades as % in most districts. At least for now.
I like your ideas… I’ve mainly been playing with trying to assign % to numbers based on my rubric, but I like the functions you’ve come up with. Thanx for posting this.

2. My district did not require percentages, just letter grades.

As for those, they were assigned by negotiation between the student and me. (More info here.)

I think that a “grail” of conversion between one system and another would comfort a lot of souls, but what’s the point? If you’re just going to use the same system of averages and percentages, you haven’t changed anything at all except the name (standards-based). In the meantime, you’ve created a lot of unnecessary statistical headaches in the process.

If you absolutely have to assign a number (instead of a letter), I would still suggest spending time discussing scores with each student and asking them about where they fall on a scale from 1 – 10. They don’t have to think about it (and shouldn’t think about it) as a percent achievement…but a measure of learning. What number represents the gestalt of their understanding?

• I think that’s where we all wish we could be – not /have/ to assign numerical grades, but our districts aren’t quite there yet (which leaves me jealous that you have such a system). I still think the conversations and implementations are important though and a vital step in getting ourselves and even perhaps our schools and districts where we ideally want to be.

• nyates314

We do report percent as course grades, not just letter.

Like PersidaB says, I would love to use neither percent nor letter grades but just to report where students are on meeting the objectives of the course. And to require that they meet all [or at least 90% of the] standards (in flexible time) before progressing to the next class.

However, I’m not entirely convinced that averages are bad to get an overall summary of where a student is. To me, the primary value I see in standards-based grading is creating a finer distinction for me and for the students themselves as to where they need to focus and seek help. Additionally, the benefit of continuing to work until mastery on assessments instead of weighting sideline pieces like daily classwork.

But I do like your idea of conferring with each student before assigning a grade.

And I’m intrigued by an idea suggested by @mthman here: creating a grade percentage based on percent of standards met instead of averaging progress on standards.

3. If you just assign integers 1,2,3,4, then you don’t need a formula to convert them to percantages: a simple table lookup of the 4 numbers would do.

One way to help if you want to use cubics is to insist that the minimum and maximum be = 4. Try the 4 constraints:
f(1)=50
f(4)=100
f'(1)=f'(4)=0

f(x,a0,a1,a2,a3) = a0+x*(a1+x*(a2+x*a3))
fit f(x,a0,12*a3, -7.5*a3, a3)

getting
a0=70.3704
a1=44.44444
a2=27.77777
a3=-3.7037

This is fairly close to the values you wanted:
print g(1),g(2),g(2.5),g(3),g(4)
50.0
62.9629629629629
75.0
87.037037037037
100.0
but is nicely monotone between 1 and 4.

• nyates314

Nice!

4. It does look like fun. And I don’t have a solution. But it does point to some of the pathology we have to deal with, when we fit a cubic to five points in order to try to have one system make sense in the scale of another. I mean, I’ve used 90-100 is an A too, but why should it be that way? What’s magical about 90? You could make it work by changing the cut-points—or, as you suggest, try to figure out something more holistic.

Me, I’m just starting SBG, so I’m wondering how I’m gonna deal with this. Thanks for bringing it up!

• nyates314

In our district, we report percentages. For example if a student earns a 92 in a quarter, they see that 92 on their report card not an A. So cutoff is not an issue for me, unless I create my own in-class cutoffs that scale by some mathematical formula to a standard score out of 100.

I’m curious too about what other more holistic ways there are out there. I’m thinking about The Science Goddess’s idea above about setting grades not by a formula but in conference with each student, and wondering if that might work in my classroom/setting.