In Part 1 of this series, I worked through how I established my rating scale and associated grade scale using the Algebra 1 CST performance level cut score percentages as a benchmark.  I ended by revealing the resultant, preliminary end of semester grade distribution, as shown in table 1.  Each grade was determined by equally weighting a student’s assignment score and assessment score.  The preliminary grade did not reflect any changes I may make based on proximity to a cut score, specific student circumstances, or etcetera.

Table 1: Distribution for Egalitarian Grading System

After contemplating a colleague’s comments, and reflecting on what a final grade for a student meant to me, I decided to explore a variety of algorithms for determining a grade within the framework I use for rating individual student work, as shown in table 2, as well as the scale I use to convert weighted average percentage scores from the rating scales into final grades, as shown in table 3. [1]

Table 2: Individual Work Rating Scale              Table 3: Grade Reporting Scale

As a point of comparison, the grade distribution in table 1 is based on weighting the average percentage score for the semester of all assignment ratings by 50% as well as the average percentage score of all assessment ratings by 50%.  I refer to this as the “Egalitarian Grading System” since every student’s grade depends equally upon their assignment and assessment score.  The upside for students with this approach typically inures to the benefit of students who do not fare as well on assessments, but maximize their assignment grade, especially since I rate assignments on completion only.  The downside falls mostly upon students who score high on assessments, but for a variety of reasons, do not rate as high on assignments.

To compensate for the downside in the Egalitarian distribution, the following grading scale assigns a student the highest of the weighted average score or the average assessment score.  In this way, the upside discussed above continues, while the downside is removed.  I named this the Progressive grading system, as shown in Table 4.

Table 4: Distribution for Progressive Grading System

Note that the number of C and above grades increased by six-percentage points with this system.

After creating the Progressive grading system, I wondered how the grade distribution might shift if the final grade was the highest of a student’s weighted average score, their average assessment score, or their average assignment score.  Table 5 shows the distribution for what I call the Ultra Progressive grading system.

Table 5: Distribution for Ultra Progressive Grading System

With this system, note that the percentage of students receiving a C or higher reaches 69%, much higher than the Egalitarian system, which is 58%, and five percentage points higher than the Progressive system, which is 64%.  However, I was not comfortable basing a student’s grade solely on their average assignment score, especially in cases where their assessment score significantly differed from their assignment score.  While this system increases the percentage of students receiving C or higher grades, it provides a grade to students that does not align with their performance level on the content standards, which I believe is a primary purpose for the grade.

Moving back to the Progressive system, and building upon it, I created the Super Progressive grading system, where a student’s score on the final exam, which is cumulative, is included in the decision criteria for their final grade.  If a student fared better on the final exam than either their weighted average score or their average assessment score, they received their final exam score.  I felt this was the most fair system to implement for my students.  It definitely benefited one student who otherwise would have failed the course. It also materially affected the final ratings of one-third of the class in a beneficial way.  Table 6 shows the Super Progressive grading scale’s grade distribution.

Table 6: Distribution for Super Progressive Grading System

While the Super Progressive grading system reduces the overall number of students who receive a C or higher, it ends up lowering the number of students who otherwise would have received an F.  Of course, I could create a Super Ultra Progressive grading system to deliver both benefits, however, it would keep the problem I discuss above with the Ultra Progressive grading system, notably that a student’s grade does not align with their performance level on the content standards.

Tiring of the lengthy adjectives defining the various grading systems, I named the system I ultimately selected the Final grading system, which consisted of taking the Super Progressive grading system and making manual increases to various student grades mostly based on proximity to a grade cut score.  Table 7 shows the grade distribution for the final grading system.

Table 7: Distribution for Final Grading System

Note that the combined percentage for students receiving a C or higher is the greatest yet at 71%, while the number of students receiving an F dropped to the lowest level of any system.  Eight students, who otherwise would have received a failing grade, are able to receive credit for their first semester of algebra 1.  To a one, they demonstrated to me that they possessed a ‘below basic,’ but passing, level of understanding of the algebra 1 content standards.  For this, they deserve to receive credit for the semester.

To what extent these eight students are able to pass the second semester is yet to be seen.  Many of them have behavioral issues, which interfere with our classroom learning environment.  At the same time, this is the primary chance in their life to learn this material, and I plan to help them do so, in spite of themselves.  Hopefully, in time, they will recognize how this benefitted them.

As you can see, grade distributions vary considerably in the various grading systems I explored, most of which varied in progressiveness.  At the other end of the spectrum, a grading system based solely on assessment, yields the distribution shown in table 8.

Table 8: Distribution for an Assessment Only Grading System

This system delivers the lowest overall percentage of students receiving a C or higher grade, and hence the highest percentage of students receiving a D or F.  As such, it is the most punitive of all systems, in my opinion, and unworthy of implementation in any high school today.

What is not clear to me, as of this writing, is the extent to which assignment scores, participation scores, and assessment scores factored into grades over the years.  There is a dearth of information on specific grade distributions at any level.  I was able to find grade distributions from the year 1934 and 1982 for comparative purposes.  [2]

The following data were taken from a JSTOR site for the article written by Norman E. Rutt, originally published in National Mathematics Magazine in 1943.  I placed each data set into the same tabular format as used above for ease in comparing the distributions.  Table 9 and table 10 contain this data.

Note that the grouped data for C grades and above, and for D and F grades, in table 9 matches exactly that of table 6, the Super Progressive grading system.  This is entirely coincidental, however, of interest for future discussion.  Mr. Rutt’s intent in publishing his data was to explore to what extent grades follow a normal distribution.  I have not assessed his data, or mine, for normality.  My data were not normalized.

Table 9: Grade Distribution from 1943 (near normal distribution)

Table 10: Grade Distribution from 1943

The data in table 10 do not seem to follow a normal distribution.  Surprisingly, the data in this table also match the grouped data for C grades and above, and for D and F grades, in table 7, the final grading system.

#### 1982 Seniors’ Distribution of Grades in Mathematics

Table 11 shows data released by the National Center for Education Statistics (“NCES”) in 1984 estimating seniors’ grades in mathematics in 1982.  Somewhat of note, I graduated from high school in 1982; however, I did not take any mathematics courses my senior year, not that I would impact data from all graduating seniors in the U.S.

Table 11: Distribution of Mathematics Grades for Seniors in 1982

These data show the most favorable distribution of C and above grades versus D and F grades.  Why this is the case is unknown.  It could be due to the differences in student socioeconomic and other demographic status at that time.  The extent to which this data exhibits normality has not been determined either, at least from my first read of the NCES bulletin.

### Conclusion

The results of the final grading system leave me feeling quite satisfied.  In fact, I was elated last night when I settled upon using it.  I find it to be the most fair and justified system for assigning final grades to students.  I believe they will feel similarly.  I have not explored how my AP Calculus students grades will fare in this system.  However, due to its very nature, students should receive a grade reflective of their performance level with the content, which is the primary goal for assigning grades from my perspective.

[1] Explaining how table 2 and table 3 relate to each other in any detail will take another post.  Suffice it to say that individual assignments are scored using a 0-100% scale which are then reported to students online using the rating scale in table 2.  Once in the online system, cumulative ratings are converted to percentages where the system uses table 3 as a lookup table to convert those percentages to the A-F grades required by the district.

[2] It is not clear to me why there is so little data published on grade distributions, since it can only benefit our educational system if we publish and explore data such as this to help develop the most effective systems possible for educating our nation’s future.  There is no personally identifiable data, either directly or indirectly, associated with such data, although I can understand why many might be reluctant to share data like this, or even have the time to do so if they wanted to share.

## About Dave aka Mr. Math Teacher

Independent consultant and junior college adjunct instructor. Former secondary math teacher who taught math intervention, algebra 1, geometry, accelerated algebra 2, precalculus, honors precalculus, AP Calculus AB, and AP Statistics. Prior to teaching, I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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### 5 Responses to Grappling with Grading: Part 2 – Grading Scales and Distributions

1. Cal says:

Okay, first up: I am BEAMING that you used the method you did, finally. Well done. But then, you know my thoughts on that.

Second up: You took a long route to get to what I would have advised in the first place.

1) Test average highest. If they’ve been consistent throughout on this, and it’s their highest average, this is their grade.

2) Test average plus effort (classwork, whatever) highest. If they’ve been consistent throughout and classwork boosts them on the plus (that is, a C+ moves them to a B-), this is their grade.

This takes care of 95% of your students.

3) Start strong, fall off OR start weak, end with effort: look at their final and give them this grade.

4) Finally, take one last look at your D- and F students and ask yourself, finally, if they demonstrated knowledge of the fundamentals (as you did).

This is a lot less work than whatever you did, I’m pretty sure!

You seem more interested in grade distribution than I am. I am always concerned about failing too many students, obviously, but ultimately, each individual earns their grade. We have far too wide a range in the classroom to worry too much about distributions.

Incidentally, I’ve been taking a look at my finals (Geometry and Algebra II). At the beginning of break, I was leaning towards creating a special scale for my students. But on second thought, I decided that I would use my usual multiple choice curve (15 points instead of 10). My Geometry students did very well on this–much better than I thought. A few annoying Fs from kids who phoned it in, but a lot of my consistent D students got a solid D or high F on the test, which means they retained their knowledge throughout the year. My Algebra II results were much less attractive–way too many Fs. Part of that was, I think, due to the fact that the test wasn’t well-designed–I was unhappy with it and there were too many typos. Not all of it, though. Some distressing answers on linear equations. Still, for the most part, in both courses, the kids who needed a good final turned one in. The kids who did badly on their own relative terms might have annoyed me, but it didn’t affect their grade.

In Algebra II, I’m able to hold to my word that, if you showed up in class and worked, you passed. However, I’m going to talk to my students and tell them that a D grade should be interpreted as the functional equivalent of an F, and remind them of the importance of placement tests and the like.

Oh, and I got your email. Most of the material I used is in books, so let’s set up time to meet on Monday. I have a few handouts I built that you’ll find useful. But mostly it’s an approach. Think of it this way: they should always have something in front of them that’s achievable without too much effort.

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2. Hey Cal.

While you may have mentioned your process to me before, I needed to go step-by-step to get a firsthand feel for how my rating scale and weighted averages interacted, as well as to ponder the import of assessments versus assignments as well as a cumulative final – call this the neophyte teacher syndrome.

Ending up where I did was not intuitively obvious, for me, given my initial belief that a grade was determined solely by whatever percentage resulted from a weighted average, with some minor tweaking after the fact. I did not envision my final grading system as the end-game since a part of me felt aghast at letting one positive event supplant a series of fails, although the other, more dominant part of me resonated with this approach once it smacked me in the head. As a “last of the baby boomers” child, with one parent ultra conservative, and the other much more progressive, I wrestle with what might seem more clear to one more purely aligned with either end of the philosophical spectrum. In the end, the progressive in me usually triumphs, as fairness is one of my strongest character strengths. See https://mathequality.wordpress.com/2011/02/12/character-strengths-teaching/ for more on why I make this claim.

BTW, I have considerably less experience on this side of the educational world as you, so my preconceived ideas of how things actually work need to be shattered through these arduous experiences. No pain, no gain, they say. I believe I am the wiser for the experience, and more importantly, I feel as if I am serving my students, and society, in the best way I know possible. I really appreciate our professional discourse, too. It helps me improve as a first year teacher.

Dave

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3. Cal says:

I have noticed that about you, that you need to mull it and test things out and run it through a million data checks. I try to respect it–no, really! Many people think I’m averse to comments or suggestions because I instantly toss them up verbally and shoot down what I think is wrong with it. But inside, I’m testing it out and mulling it, so I hate it when people think I’m ignoring their ideas. Likewise, I don’t want to make you feel as if you should have listened to me to start with, or anything.

At the same time, I see you working incredibly hard. If I see you overthinking, I’m always going to push back a bit, simply because there are all sorts of things about teaching that are worth the hard work, and some things that ultimately aren’t. You will always have to make that call for yourself; I’m just speaking up to give you another perspective.

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• Push away, my friend. Its within that tension that true discoveries are made…

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