Portfolio Instructions – MTH 350, Winter 2022

What is the Portfolio and how will it be used?

A portfolio is a collection of work that gives information about the person doing the work. Artists and architects, for example, have portfolios that show the range and scope of what they do, and by looking through their portfolios, you get a sense of who this person is, how they have grown as a professional, what they have produced, and what they are capable of producing.

That's the spirit intended for the MTH 350 Portfolio. In your portfolio, you will give a curated collection of your best work along with responses to some essay prompts that tells the story of your growth from the first day of classes all the way to the end. Your goal is to argue, convincingly and with concrete evidence, that you have met the criteria for both content mastery and class engagement that goes with a particular grade. I will read your portfolio carefully, and if you make a convincing case for a grade, that's the grade you'll receive.

Key information

Format: You'll put all of your work –- both the artifacts of past work and the responses to the writing prompts –- into a single PDF that you will upload to Blackboard in a specific assignment area that will be set up later. Here is a

$LATEX$ template that you can use. The template is read-only to prevent accidental changes; just copy/paste all the source code into a new

$LATEX$ document in your Overleaf account or wherever you work on

$LATEX$ documents.

Due date: The portfolio is due at the end of your section's Final Exam period as shown on the Class Calendar. No deadline extensions are available for the Portfolio.

What to include

Your portfolio needs to contain all of the following:

• Table of contents. List all the items in the portfolio in the order in which they appear. Include the number and name of each proof, or a title for other items (for example, "Presentation of Daily Prep Problem 2 on March 17").
• Essay responses. Write brief responses to each of the essay prompts listed below under Essays. Put each essay response on its own page. (All of the "Lightning Round" responses should go on the same page, however.)
• Final grade reflection. This also goes on its own page (or pages). See below for details.
• Supporting artifacts. After stating what grade in the course you believe you've earned and reflected on your reasoning, carefully select items that demonstrate concretely and directly that you have met the criteria for that grade. Your homework solutions and class journal submissions should be a primary source of these, but you can also include other items. For example, if you gave a presentation that helps show how you've met some of the criteria for a grade, include it (maybe along with your notes on it). Anything that supports your argument for a grade can be submitted. But please keep these focused, and be selective. If something doesn't directly address one of the grade criteria, you don't need to include it and probably shouldn't.

Revisions

You can revise any homework or other artifact one more time before including it in the portfolio. If you do so, please note:

• Do NOT submit revised Homework solutions to Blackboard under the Homework assignment area. Just put the revision in the Portfolio.
• In the Portfolio, to help me understand what you are turning in, make sure if you submit a revision to indicate this somehow. For example in the narrative for your homework solution, say "This is an additional revision done for the Portfolio."
• You won't get any additional feedback for revisions done for the Portfolio. If you want feedback on a Homework solution, submit it as an ordinary homework revision (subject to the one-revision-per-week rule). Please note, as described in the Syllabus, I do not pre-validate work before it's submitted so requests to review revisions before going in the Portfolio will be declined, unless you submit them as ordinary homework revisions.

Essays

In your Portfolio, include a short essay on each of the following prompts. Label each essay clearly by name, and put each on a separate page.

• Growth as a mathematician: Describe how you have improved as a practitioner and student of mathematics this semester through your work in the course. Focus specifically on algebraic ideas and the methods we employed in learning them. Don't just give a laundry list of topics that you learned (those are in the syllabus, so we know them already). Focus instead on a small number of specific areas of growth for you that happened in MTH 350. Identify at least one artifact in your portfolio that illustrates each area of growth. Explain in this essay how this artifact shows growth in the way you are describing.
• The power of abstraction: As we've said in the class, MTH 350 is sometimes called "abstract algebra". In this essay, give an explanation of what that term "abstract" means in relation to mathematics and describe some places where you encountered abstraction and understood its importance. Include: What is the definition of "abstract" or "abstraction"? What does abstraction look like in math? What does it allow you to do? Why is it important? Where did you encounter it, and what were some "a-ha" moments you had about abstraction? (For the latter, tie each a-ha moment to an artifact in the portfolio.)
• Lightning round: Answer each of the following questions with one sentence including a brief explanation.
  • Finish the sentence with another student’s name: “If [Student] doesn’t earn an A, then nobody should, because…” Explain your choice and your reasoning.
  • What was the most difficult part of the class for you? List a general topic, a specific problem, a kind of work we did, etc. and explain why.
  • What was the easiest part of the class for you? Explain why.
  • What part of the class surprised or interested you the most? Explain why.
  • Give one piece of advice to a student just beginning a semester of MTH 350 that will help them be successful in learning.

Final grade reflection

This component is a self-evaluation of your performance in the class. Your goal is to state clearly what grade you think you have earned in the course, and convincingly argue that you have met the criteria for that grade. Please be thoughtful, honest, and reflective, but also brief and focused –- spend no more than 2 pages on this part. Structure your response as follows:

1. What grade did you earn? State it clearly in one sentence. You can include plus/minus modifiers; see below about those.
2. How did you meet that grade's requirements? The requirements for grades are found in the How Do I Earn a Grade? section of the syllabus. Review the criteria carefully. Then, explain how you have satisfied the criteria for the grade you are saying you've earned. Be specific and thorough. List each criterion you have met and how you know you've met it. Refer to specific artifacts or examples in the Portfolio that support your case. Again, keep it brief and focused on concrete evidence that the criteria have been met.
3. What else? There may be some other criteria that figures into your grade request that isn't explictly listed in the syllabus. If so, state the criteria you're thinking of, explain why they are useful criteria for your grade determination, and then explain how well you met those criteria (and provide evidence in the artifacts, if applicable). As with the syllabus criteria, be focused, specific, and thorough –- and refer to specific artifacts that provide evidence.

Plus/minus grades: You may argue to add a "+" or "-" modifier to your grade. Generally speaking a "+" added to a letter grade means you've met all the criteria for the letter grade along with a significant amount, but not all, of the criteria for the next letter grade up. And a "-" means that you have met the requirements for a letter grade but only in a way that you believe is a bare minimum, or you've met almost all the requirements for the letter grade except for a small number of minor criteria.

About time and effort

As your instructor, I understand and respect that you have spent a great deal of time and effort on this course and will probably continue to do so until the end of the semester. It's possible that MTH 350 has been the most challenging course on your schedule this semester, perhaps in your entire college career so far. I want you to know that I see you, I respect your efforts, and it's extremely fulfilling to see you respond to the challenges and grow.

I also want to be clear that while the time and effort spent on the class are vitally important, they are not explicitly part of the grade criteria you find in the syllabus. So, when you make the case for your grade in the portfolio, I respectfully ask that you keep your argument focused on concrete evidence of understanding of algebraic ideas and engagement with the class.

You are free to discuss your investment in time and effort in the essay questions if you like, or apart from the portfolio if that makes more sense. But when making the case for your grade, the focus needs to be on the results of your investment of time and effort. Did your efforts result in several exemplary solutions through the semester? If so, put a subset of them in the portfolio. Did the time you spend on Daily Prep result in consistent volunteering for presentations? If so, document when you volunteered.

Questions you might have

Click here for the Padlet that contains questions about end-of-semester items.

Q: What if you disagree with our final grade reflection?

A: There are two ways this might happen.

1. You ask for a grade that is higher than the evidence supports. That is, you grade yourself too highly, for example you believe you earned an A in the course, but the evidence doesn't support it. In this case I will email you to request a meeting with you to discuss the disagreement. I'll tell you what grade I would have given you based on the evidence that you provided in the portfolio, and then give you the chance to provide additional evidence and explanations for your grade request. In our meeting, if you still think your grade request is correct, you'll have the chance to walk me through your materials and reasoning. Afterwards, I might agree with you. But if we still disagree, we will try to come to an agreement on the highest grade that the evidence supports. I have faith that we'll be able to come to a mutually-agreeable result. But, if we can't, I will assign the grade that in my estimation is the highest grade supported by the evidence, and my determination will be final. However, again, I will exhaust all other options before simply overriding your grade determination.
2. You ask for a grade that was lower than your actual performance indicates. That is, you "lowball" yourself, for example you say you earned a B when in fact I think you earned an A-. In this case, I will award you the grade I think you earned, and then just inform you that I've done so (and why).

In my experience, the second situation happens significantly more frequently than the first.

Q: Is there a limit on the number of artifacts I can include to support my grade?

A: There is no strict limit, but please be selective. Quality, not quantity is what we are after.We do not want 100-page portfolios! Typically you should be able to give a complete portfolio in fewer than 25 pages with judicious selection of homework solutions and other work. A handful of homework solutions and class journal submissions that clearly indicate your attainment of the grade criteria is just as good as, if not better than, a massive binder of mediocre examples.

Q: How specific or detailed do I need to be on some of the criteria? For example do I need to list all the dates when I made a supportive comment in a presentation?

A: This isn't filing your taxes, so a precise down-to-the-minute accounting of your work isn't necessary. Details and your reasoning are more important than precision and completeness here. For example if you made a handful of supportive comments in presentations, pick a few and tell me about them. Whose presentation was it, and what were they doing? What did you say? Why was that comment an example of "helpful feedback"? (Do you know that it was helpful?) Remember the purpose is to make a convincing case that you've earned a particular grade.

Q: Would someone actually make a case that they earned a "D" in the class?

A: Yes, if the person is honest and there's not enough evidence to make the case for a "C". If you're in that situation, be honest and don't overreach just because the consequences are significant.

Q: How long should the essays be?

A: You should put serious effort into each of the essays and give thorough, detailed, and thoughtful responses –- without going overboard and writing more than you need. There is no hard word or character count here, but if you are writing less than half a page using ordinary font and line spacing, then you probably need to spend more time carefully reflecting on the question; if you are writing more than two pages for a single essay, you might consider editing it down for clarity. As always, put yourself in the shoes of your readers and then use your judgment as a writer.

(From the padlet) One of the requirements is a deep understanding of polynomials. We have not done any homework problems on polynomials, and I don't see any class journals problems involving polynomials. Other than potential presentations, what else are we able to use as evidence here?

A: That language in the syllabus was written at the beginning of the semester. Since then, we have significantly scaled back the amount of coverage in the class to reflect the need for review and reteaching. Since polynomials were at the end of the course already, we ended up not having a lot of major assessments on these. Therefore I've just updated the language in the syllabus to say:

Demonstrate deep understanding of these topics: Integer divisibility and prime numbers, equivalence relations, the integers modulo n, fields, rings, properties of rings/fields and ring/field elements. Items related to polynomials are encouraged but not required.

So if you've got something to share about polynomials – a problem from the list, or a presentation, etc. – it would be great to include it. But if you don't, just make sure you're presenting good evidence for the others.

We'll add more questions and answers in this space as they come up.