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How Assessment and Grading Work in MTH 325

As with any academic course you take, you will be asked to do various kinds of work in MTH 325 and submit them to me for assessment. I then apply my best professional judgment to your work and assess its quality, and eventually you will receive a course grade based on a composite of those professional judgments. This is a big responsibility for me. In designing this course, it is imperative that I make sure the assessment system you have in MTH 325 satsfies some important requirements:

  1. Your grade in the course should reflect the amount of mastery you have attained on the course material and the work that you put into the course.
  2. Your grade in the course should send a clear message about what you have and have not mastered in the course. Two students with the same course grade should have the same level of content mastery.
  3. Your course grade should not be a "false negative". That is, you should not receive a low course grade when you have in fact demonstrated significant mastery of the material.
  4. At the same time your course grade should not be a "false positive". That is, you should not receive a high course grade when you have not in fact demonstrated significant mastery of the material.
  5. Finally, the course grade should reflect your progress over time as a learner and your mastery of the material that you've attained at the end of the semester. If you are given the opportunity to demonstrate mastery of a concept but for whatever reason fail to do so, you should receive opportunities to try again, rather than have your course grade permanently lowered by one poor performance on an assessment early on in the course.

It is my determination that the traditional system of accumulating and manipulating points based on homework and timed testing does not meet any of the above criteria consistently. Therefore we will not be using the traditional points-based grading system in this class. Instead, we will be using a system of assessment and grading that is based on meeting learning benchmarks throughout the semester. In the remainder of this document, we will describe this system and try to be very clear about how it works.

This system is closely related to "competency-based" education used by numerous higher education institutions (including Duke University and many online institutions), corporations, and certification agencies throughout the world as well as "standards-based grading" used in many universities and schools today (including my own kids' schools here in Allendale). It is designed to provide a true picture of your mastery of the course material, allow you to learn from your mistakes and grow as a learner, and put you firmly in control of the grade you receive in the course.

Types of assessed work

During the semester, you will be turning in work quite frequently. Some of these are formative assessments designed to provide quick turnaround feedback for your progress while others are summative assessments for seeing how well you have mastered a body of content once that unit is complete. The kinds of work you will do are Guided Practice, Concept Quizzes, Learning Modules, Timed Modules, and an Application Project.

Guided Practice

The structure of our course is such that you will be asked to master certain foundational learning outcomes prior to coming to class. This will free up significant time for work on more complex and difficult ideas, while we are together as a group. Please see the handout "Student and Instructor Expectations in MTH 325" for more information. Your individual pre-class preparation is therefore crucial for your success in the course. To help you prepare, you will receive frequent pre-class assignments called Guided Practice assignments.

Guided Practice assignments are posted to the course website around 10 days prior to class coverage on a particular topic and are mapped onto that topic. For example, we are planning to work on planar graphs (Section 9.7 in Rosen) on February xxx; around January xxx you will find the Guided Practice for this topic on the website. Each Guided Practice consist of an overview of the topic, learning objectives to let you know what you need to learn before coming to class and what we will work on during class, a reading assignment and other helping content such as video or applets to help you learn the material, and some short exercises on the basic learning objectives. You will submit these through an online form. They are always due no later than one hour before your section's meeting time on the day when we discuss that topic. (For example, the Guided Practice on planar graphs is due one hour before class time on February xxx.)

I use the results of your Guided Practice to look for patterns in student understanding of the material before class starts, which allows me to catch misconceptions or misunderstandings before you arrive, devise some examples or further work to address them and make them a high priority when we meet.

Guided Practice assignments are graded on a scale of 0, 1, or 2 based on completeness and effort only. Correctness is not one of the grading criteria! So you are not penalized for mistakes; you are only penalized for non-submissions, or for submitting work that does not demonstrate a good-faith effort. For example, submitting "I don't know" or leaving an exercise blank will result in not receiving full credit.

Concept Quizzes

Periodically in class (I am planning roughly once a week), we will have short timed quizzes over basic information relevant to the material we've just covered. This information includes definitions of terms, statements of mathematical theorems, and very simple calculations. These Concept Quizzes are typically worth 12 points and will often be fully objective, using multiple choice, true/false, or fill-in-the-blank questions that do not involve partial credit. These will be announced in advance, and you'll receive a list of the topics to be included on each of these in advance.

Learning Modules

The heart of the course consists of a series of Learning Modules that you will work individually in untimed settings -- mostly outside of class, but every so often we may set aside a class meeting to work on these.

For each of the three major concepts of MTH 325 -- relations, graphs, and trees -- there are three modules:

  • Relations 1: Representation and Properties (corresponds to Rosen Sections 8.1 and 8.3)
  • Relations 2: Closures (Section 8.4)
  • Relations 3: Equivalence Relations and Partially Ordered Sets (Sections 8.5 and 8.6)
  • Graphs 1: Representation and Properties (Sections 9.1--9.3)
  • Graphs 2: Connectivity, Paths, and Circuits (Sections 9.4 and 9.5)
  • Graphs 3: Shortest paths, Planarity, and Coloring (Sections 9.6--9.8)
  • Trees 1: Properties (Section 10.1)
  • Trees 2: Applications (Sections 10.2 and 10.3)
  • Trees 3: Spanning Trees (Sections 10.4 and 10.5)

Please see the MTH 325 Learning Outcomes document for a micro-scale list of learning objectives for each of these topics. Additionally, each module has two Levels:

  • Level 1 modules focus on computation, basic conceptual reasoning, and applications.
  • Level 2 modules focus on more advanced concepts involving application, analysis, programming, and mathematical proof.

So there are (three topics) x (three modules) x (two levels) = 18 content-based learning modules in all. Nine of these modules are what you might call "basic" (Level 1) while the other nine are "advanced" (Level 2).

Additionally, there are two other modules that assess general skills for the course:

  • Getting Started: This module, which will be due in the second week of class, will ask you to respond to questions about the syllabus, this grading system document, and other course information; write some responses to articles about learning and about the use of mathematics in computing; and set goals for yourself for the semester, including the grade you'd like to earn.
  • Tech Competency: This module will ask you to demonstrate competency in using basic course technology such as your GVSU email account, our discussion board, the course website, SageMath Cloud, and the Sage computer algebra system.

Timed Modules

In addition to the learning modules you will do outside of class, you will also be asked to work nine sets of problems through timed in-class work. These nine timed modules correspond roughly to the nine Level 1 learning modules described above. We will set aside four 50-minute sessions in the class for you to work these modules.

Unlike a traditional timed test, however, you get to choose which timed modules you'd like to work on in any given timed assessment session. Also, if you do not successfully complete a timed module in one assessment session, you may try it again in a subsequent assessment session.

For example: For the first timed assessment session, Alice decides she'd like to try the timed modules on Representation and Properties, and on Equivalence Relations/Partially Ordered Sets. She's worked a little on the (untimed) module on Closure and doesn't feel confident enough yet that she'd be able to pass it, so she's going to save that one until the next timed assessment session in a few weeks. After the session is over and the work is graded, she finds she has passed the Representation and Properties timed module but not passed the Equivalence Relations/Partially Ordered Sets timed module. So, at the next session, she plans on retaking the module she didn't pass, in addition to the module she didn't try (Closure), and a couple of new modules that have been discussed in the class since then.

Alice can continue taking and re-taking timed modules all the way up until the final exam session (see below). Each timed session (other than the final exam session) lasts only 50 minutes, so the idea is to choose the problems you feel confident about working and make sure you finish them successfully, and save the others for later. In this way, an early poor performance on one topic does not doom your grade. You can work some more and try again at a later date under similar conditions. However, you do need to make steady progress, or else you will have too many timed modules to complete for the amount of time in a session.

Every student must make an attempt of at least one timed module in each session unless that student has successfully completed all the ones required for the grade the student chooses. (See below for information about course grades.)

Application Project

The Application Project is a large-scale project in which students will work in teams to solve a problem or provide a service that involves relations, graphs, and/or trees. The application project culminates in a public poster presentation open to the entire campus community during the last week of classes. Much more information on the Application Project will come your way around week 3.

How Individual Items are Assessed

In the description of the assessed work for the course, you saw references to "passing" or "successfully completing" modules. We need to explain what we mean by this.

Your work on Learning Modules, Timed Modules, and the Application Project is graded "pass" or "does not pass", using a two-level rubric based on a variety of professional standards and specifications. There is no partial credit awarded. Instead, you will be responsible for ensuring that your work meets the standards that will be set out for you.

If this sounds intimidating, it's really not. I will be providing you with highly detailed descriptions of what "passing" work looks like, including examples of passing work and non-passing work, and we will also spend time early in the semester on training how to distinguish passing from non-passing work using these examples. Also, please keep in mind "passing" does not mean "perfect". You will be allowed some mistakes on your work provided they are not crucial. Generally speaking, a "pass" grade is what would normally be assigned a "B" under a traditional grading system -- not necessarily perfect but demonstrating a clear mastery of the concepts at hand. Finally, you will be getting extensive feedback on your work intended to help you reach "passing" work consistently.

You will also have a safety net that will provide you with limited opportunities for revisions or late work, as well as an open-door policy for contesting no-pass grades. See "Safety Nets" below.

How your semester grade will be determined

The grade you earn in this class is entirely up to you. To earn a particular grade, all you need to do is complete a certain number of tasks associated to that grade before the end of the cousre. There is no point accumulation, no statistical calculations, and no uncertainty as to where you stand with your grade at any time.

How to earn basic letter grades

The scheme below shows how to earn an A, B, C, D, or F in the course.

  • To earn an F: Fail to meet the requirements of a D.
  • To earn a D:
    • Pass the Getting Started and Tech Competency modules.
    • Complete 50% of Guided Practice assignments with a grade of "2"; AND
    • Pass 5 (out of nine) Level 1 learning modules; AND
    • Earn 75% of possible credit on Concept Quizzes.
  • To earn a C:
    • Complete 75% of Guided Practice assignments with a grade of "2"; AND
    • Earn 75% of possible credit on Concept Quizzes; AND
    • Pass all nine Level 1 learning modules; AND
    • Pass 6 (out of nine) Timed Modules.
  • To earn a B, complete all the requirements for a C as well as the following:
    • Pass four Level 2 learning modules; AND
    • Pass 2 additional Timed Modules (that is, pass 8/9 of the timed modules); AND
    • Complete an Application Project with passing marks.
  • To earn an A, complete all the requirements for a B as well as the following:
    • Pass four additional Level 2 learning modules (for a total of 8 out of 9).

Note that under this system, each student is allowed to choose the grade that is the most realistic and desirable goal for him or her, and only do the work that is associated with that grade. In other words, each student opts into the grade and the workload that he or she wishes to take on. I encourage each of you to aim high. But if you have given it careful consideration and all you want in the course is a C, then you are an adult and I will respect your wishes -- and the above description shows exactly what you need to do. Once you have met all the criteria in the description for a C, you have earned a "C" -- no further work will lower that grade, although there are plenty of opportunities to raise it through additional work.

In particular, note that the Application Project is only required of those students choosing to pursue a grade of "B" or higher in the course. This way, the Application Project will be done only by the most highly-motivated students in the course.

Here is another way to look at the grading system:

  • Students who earn an A in the class will have done the following:
    • Passed the Getting Started and Tech Competency modules.
    • Completed at least 75% of Guided Practice with a grade of 2/2.
    • Earned at least 75% of possible credit on Concept Quizzes.
    • Passed all nine Level 1 learning modules.
    • Passed 8/9 Level 2 learning modules.
    • Passed all nine timed modules.
    • Completed the Application Project with a passing mark.
  • Students who earn a B in the class will have done the following:
    • Passed the Getting Started and Tech Competency modules.
    • Completed at least 75% of Guided Practice with a grade of 2/2.
    • Earned at least 75% of possible credit on Concept Quizzes.
    • Passed all nine Level 1 learning modules.
    • Passed 4/9 Level 2 learning modules.
    • Passed 8/9 timed modules.
    • Completed the Application Project with a passing mark.
  • Students who earn a C in the class will have done the following:
    • Passed the Getting Started and Tech Competency modules.
    • Completed at least 75% of Guided Practice with a grade of 2/2.
    • Earned at least 75% of possible credit on Concept Quizzes.
    • Passed all nine Level 1 learning modules.
    • Passed 6/9 timed modules.
  • Students who earn a D in the class will have done the following:
    • Passed the Getting Started and Tech Competency modules.
    • Completed at least 50% of Guided Practice with a grade of 2/2.
    • Earned at least 75% of possible credit on Concept Quizzes.
    • Passed 5/9 Level 1 learning modules.
  • If a student earns an F in the course, it means at least one of the following:
    • The student did not pass the Getting Started module, OR
    • The student did not pass the Tech Competency module, OR
    • The student passed four or fewer of the nine Level 1 learning modules, OR
    • The student earned less than 75% of possible credit on Concept Quizzes, OR
    • The student completed fewer than 50% of the Guided Practice with a 2/2 score.

How to earn plus/minus grades

Students opting for grades of B, C, or D can attain plus/minus grades as follows:

  • To earn a D+: In addition to completing the requirements for a D, either earn 90% on Concept Quizzes or pass three timed modules.
  • To earn a C-: In addition to completing the requirements for a D, earn 90% on Concept Quizzes and pass three timed modules.
  • To earn a C+: In addition to completing the requirements for a C, earn 90% on Concept Quizzes or pass eight timed modules.
  • To earn a B-: In addition to completing the requirements for a C, earn 90% on Concept Quizzes and pass eight timed modules and pass one Level 2 learning module.
  • To earn a B+: In addition to completing the requirements for a B, earn 90% on Concept Quizzes or pass six Level 2 learning modules.
  • To earn an A-: In addition to completing the requirements for a B, earn 90% on Concept Quizzes and pass six Level 2 learning modules.

Note that the grades of A+ and D- are not given at GVSU.

Safety Nets

This system of grading emphasizes high academic standards and rigor. Going along with that, you have several opportunities to correct work that is not of sufficient quality.

The final exam period

Your section's final exam period, which is a 110-minute block of time, will not contain a traditional final exam. Instead, this block of time will be used for taking any timed modules that have not been passed prior to the end of classes, as well as your lowest three Concept Quizzes.

If you are happy with your timed module and Concept Quiz situation coming into the final exam, you are not required to attend. For example, a student aiming for a "B" in the class who has passed 8 out of 9 timed modules and has earned 82% of the possible credit on quizzes (and who has completed all the other requirements for a B) can opt out of the final exam period. Or, that student can show up and attempt the remaining timed module and revise her lowest three quiz scores and possibly raise her grade if successful.

The token system

Each student also has limited opportunities for revision and mistake-correction through the use of tokens. Each student begins the semester with five tokens. A student may cash in a token in exchange for one of the following:

  • A do-over on a learning module (not a timed module, since those already have revision built in); or
  • A 24-hour extension on a learning module; or
  • A makeup timed assessment session, provided it is done within 24 hours of the original session. (This does not apply to the final exam period.)

I will remain open to suggestions for other uses of tokens -- I'm sure you will think of some creative ways to cash them in.

Grade appeals

If you have been given a grade that you think your work doesn't deserve -- in particular a non-passing grade on a learning module -- then you have the right to schedule an appointment with me to discuss it. I will ask you to provide a sound argument for why your work really does meet the specifications for acceptable work and will listen to this argument, then give my best professional judgment on your case. Coming into the office to appeal such a grade will not cost any tokens, but please do not abuse this for frivolous grade disputes when no serious case can be made.


This system of grading is quite different than what you may be used to, but I believe it is highly beneficial for you, for the following reasons:

  • It puts you firmly in control of the amount of work and level of mastery that you wish to take on. You opt into the grade you want and the level or work that you want.
  • At the same time, every student can aim for high grades and there is a clear path for how to attain them, with opportunities for revision and "productive failure".
  • Your focus in the course will now be set on mastering the concepts rather than on scoring enough points.
  • Your grade will actually mean something because each grade is mapped unambiguously to a set of learning outcomes that must be mastered in order to pass. A student cannot get a B in the class by getting 80% of everything right and hoping for a favorable statistical outcome.

I want to encourage you that if you have any questions at all about the grading system in MTH 325, please feel free to ask. I am open to making changes to the system as necessary, so please let's talk if there is something on your mind about what we are doing.

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