In my last post, I tossed out a loose taxonomy to name four different types of problems:
- Content Learning Problems
- Exploratory Problems
- Conceptual Understanding Problems
- Assessment Problems
I felt it necessary for myself. Up until now, I’d been labeling all problem equally: they’re problems! They’re tasks that are supposed to get students to learn stuff! But that implies a one-size-fits-all-ness that I don’t think is practical. The planning, time frame, facilitation, scaffolding, and – for our purposes in this post – assessment and wrap-up all look different, even if the task itself doesn’t look that different (after all, ideally we’re all using nonroutine problems with a low bar and a high ceiling regardless of whether it’s being used for formatively assessing student understanding or creating new knowledge).
It’s tough to throw out exact examples for assessment since we’re all working from different standards and tools. So I’m going to restrict it to the following universe of things to assess problems on: New Tech Network’s (where I work) most common Schoolwide Learning Outcomes (SWLOs) and the Common Core Standards of Mathematical Practice.
Now, different teachers and different schools I’ve worked with utilize these different halmarks differently. In fact, many schools have difficulty even defining many of these indicators of student learning, let alone assessing. But nevertheless, we’re trying to get a general look and feel to what a problem rubric would look like, depending on what you’re actually trying to accomplish from said problem. We’re talking broad-brush here.
Content Learning Problems
Things to assess: Oral Communication, Professionalism/Work Ethic, Make sense of problems and persevere in solving them, Look for and make use of structure, Look for and express regularity in repeated reasoning
This might just be personal preference, but I’d be wary of assessing content knowledge in a learning opportunity for a student. If we are distinguishing between learning and confirmation problems, we might want to more rigorously assess content on the latter. Another one of my favorite wrap-up activities is this quick check-up as an exit ticket.
Exploratory Problems
Things to assess: Critical Thinking, Oral Communication, Collaboration, Model with Mathematics, Construct viable arguments and critique the reasoning of others, Use appropriate tools strategically
Assuming that the time-frame is a bit longer for an exploratory problem, and that the solutions and solution routes are varying, the wrap-up could consist of a formal presentation, followed by panel-style questioning.
Conceptual Understanding Problems
Things to assess: Critical Thinking, Collaboration, Written Communication, Reason abstractly and quantitatively, Construct viable arguments and critique the reasoning of others, Look for and make use of structure, Look for and express regularity in repeated reasoning
Here, I think it makes sense to have students reflect on and communicate what they’ve learned.
Assessment Problems
Things to assess: Critical Thinking, Written Communication, Reason abstractly and quantitatively, Use appropriate tools strategically, Attend to precision
In this case, one can easily envision a rubric that assesses the items above. Assuming these tasks are a bit more individualized, a written piece – almost like the free response section of an AP exam – might make sense. I’ll leave it up to the reader’s discretion whether or not to allot numerical point values.
============================
With these self-recommendations in hand, we can more easily (hopefully!) pick and chose what would go in a rubric and where, if a rubric is one of the tools in your toolbox.
Again, the idea is to make things easier, not more complex. And to better target outcomes for each and every problem. From these recommendations we might be able to construct a loose, lean problem planning template that is directly tied to the indicators you’re trying to peg with a particular problem. Maybe even some planned facilitation and scaffolding moves as well.
Students would be asked to manipulate the sliders until they think they have the line of best fit.
Inheriting the wind; these are two of my favorite books about math
Baseball Prospectus
I never liked baseball as a kid. Maybe it’s because I wasn’t any good at it. Maybe it’s cause I never went to a professional game. Maybe it’s because it is quite boring when you watch it on TV. Then in the late 90′s the sabermetric revolution upended the stuck-in-the-50′s baseball establishment by using data to prove and disprove various myths that were pervasive in the game. From roster construction to in-game tactics, the sabermetric community was one or two steps ahead of the rest of the game. It was this data-driven analysis that served as my entry point into the game. Eventually the data-movement coalesced under the Baseball Prospectus name. Housed at baseballprospectus.com, the writers produce an annual that is my notice that Spring has arrived.
The annual contains copious amounts of raw numbers, advanced metrics, data tables, projections, as well as an approachable and easy-to-digest writing style that I blast through every March. This year was no different. This is one of my favorite books to read every year.
Measurement
Here we have another book grounded in mathematics: Paul Lockhart’s Measurement. Here we have a rich text of mathematical creativity and imagination. In fact, pretty much everything in the book is developed in the author and reader’s imagination.
The problems posed (some of which even have Lockhart’s proof to accompany them), are decidedly abstract in nature. The problems rely on ingenuity for a solution. Lockhart is more likely to use mental shape-folding than a two-column proof to describe a mathematical concept, let alone a spreadsheet of data. This is one of my favorite books.
Both of these books (in Baseball Prospectus’ case, the annual publication) are quite dear to me. They also represent two entry paths through mathematics. One uses messy numbers and data to explain why things are the way they are. The other uses clean, imaginary shapes to explain why things are the way they are in our imagination. They both feature clever, humorous, conversational writing including analogies and storytelling.
Inherit the Wind
Then there is the final scene of Inherit the Wind. After psuedo-successfully defending a high school teacher who dared to teach evolution in the classroom, the protagonist of the true-story, defense attorney Henry Drummond picks up the Bible in one hand, Darwin’s The Descent of Man in the other, and exits the courtroom. Both of these books reveal things about the nature of man, and you’d be a fool to entirely discard either.
There are context-rich ways of posing mathematical tasks. There are entirely abstract ways of posing mathematical tasks. There are interesting and engaging ways of posing problems. There are dry and uninteresting ways of posing math problems.
I hated math in high school. All the way up until my integrated Physics/Calculus class threw an old computer off the football stadium and recorded it on video and we were able to successfully approximate the gravitational constant. I’m not sure if I had been presented abstract math in more interesting ways I’d have latched on to it. I was a pretty detached kid. It’s possible, but I was also more prone to think about mid-90′s Indians baseball and Carlos Baerga’s VORP. That would have been a way for me to engage with mathematics in High School. In fact, I did engage in it when Rob Neyer would publish a new column. But that was me.
I’m not saying that every task needs to be grounded in the real world. But the number of contextual mathematical tasks that should be provided any given year is certainly greater than zero. It’s probably greater than 10. Maybe the tasks need not even be that authentic. I mean, is throwing a computer off a building a “real-world” situation? Or was it just a fun thing to do that we then did a bunch of math on? It was “real-world” in the sense that we saw it happen. We interacted with it physically, visually, inter-personally and mentally. I have not, to this day, had a professional reason to toss a computer off a stadium.
Mathematics is so wonderful and ubiquitous that anyone can have some sort of entry point into the subject. It’s too vast to be constrained to a single context or a single person’s imagination. Our access points to the subject change from person to person, from age to age. You’d be foolish to eschew context-dependent scenarios to explore mathematics. You’d be foolish to toss aside all imagination from the content. You’d be foolish not to explore any and all avenues of all ways to provide access to this remarkable subject to your diverse students. You’d be inheriting the wind.