Math Syllabus Bootcamp Part 4: Anchor Problems. A Hilbert-ian approach to curriculum mapping

In 1900, mathematician David Hilbert famously published 23 as yet unsolved math problems. The problems covered a large swath of math fields. They served as a challenge and inspiration for 20th century mathematicians. 

I propose taking that same approach to laying out your content units for the year. 

Most syllabi showcase content via unit titles:

• Unit 1: Introduction to Functions 
• Unit 2: Linear Functions
• Unit 3: Linear Equations and Inequality in 2-variables
• Etc. 

In addition to being unapproachable, unit titles keep the math hidden beneath a bushel.

Instead, considering identifying the ten (or however many) problems that well represent the content you’re covering over the course of a year. These problems represent the apex of challenge, joy, and excitement for each unit or cluster of units. Go ahead and put ‘em in the syllabus. Post them around the room (perhaps an aspiring, if troubled, janitorial service engineer will solve them when you’re not looking). Don’t keep them hidden. Give students an idea of where you’re headed. Let them know what your class is about; let them know what math is about.

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The First Four (of, let’s say, Ten) Krall Geometry Problems (most of which were initially posed by other mathematicians)

Problem 1: Pizza Delivery Problem

Problem 2: Which Triangle is More Equilateral? (from Mr. Honner)

Problem 3: Glasses (from the Shell Centre)

Problem 4: Elmo’s Microwave Travel (from Andrew Stadel)

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Not only will your ten (or however many) challenge problems offer engagement, forward momentum, and sneak-peaks into content to come, they can also serve as your Portfolio Problems for the year. 

I’ve written about, talked about, and given presentations on student portfolios as the cornerstone of an academically safe and challenging math classroom. I recently published a “how to” in NCTM’s Ear to the Ground magazine. The short version is this: students work on complex tasks throughout a school year. With revision and reflection, they’ll demonstrate their growth as a mathematician and actually yield better data than, say, unit tests or semester exams.

My recommendation is to sketch out these problems before the year even starts. Go ahead and put roughly when they’ll appear on your calendar (knowing that things will assuredly change). Create a rubric for each problem (perhaps even using a version of the DRAFT SoP rubric, which is a draft). These will be your primary conversation pieces with students, parents, colleagues, and even administrators. 

At the end of the 20th century most of the Hilbert Problems were solved, at least partially. Yet some still remain unanswered or unresolved (with a possibility that they are unanswerable). They provided guidance about what might be possible given the appropriate devotion. It might be worth communicating the same information to your students, instead of with the typical unit title. 

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The next installment for this miniseries will be its conclusion. There, we’ll put together all the concepts covered here and also discuss the nuts and bolts we’re all more familiar with in syllabi: office hours, contact information, grading, etc. You’ll also get a lovely look at my mediocre design skills as I’ll provide a sample syllabus that portends to be aesthetically pleasing.

Be sure to check out the other posts in the series in the meantime.

• Intro Post
• Part 1: Identity. Who is a mathematician?
• Part 2: Smartness. What does it mean to be a mathematician?
• Part 3. Norms. What are the expectations for quality collaborative work?
• Part 4: Anchor Problems. A Hilbertian Approach to Curriculum Mapping.
• Part 5: Putting it all together. Additional nuts & bolts and an example syllabus.