I’m wrapping up my series of posts on problem design. It was fun; sorting worthwhile problems by the Standards for Mathematical Practice has been enormously helpful in writing problems for my students every day. This idea was inspired in part by Ben Blum-Smith’s post on a language for problem design, but also from my frustration with conversations about the Common Core centering on “conceptual understanding” and “deeper learning” without defining what those terms meant. Conceptual understanding and deeper learning are certainly important however they are defined, but they’re defined differently by every math teacher who uses them. The Standards for Mathematical Practice give teachers a common language for the specific habits of mind and types of understanding students need to be proficient at mathematics.
There are tons of great resources that have been built all over the MathTwitterBlogosphere around the Practice standards, but many of them focus on rich tasks, inquiry-based instruction, 3-acts, and similar materials that teach full lessons based on these practices. While I love many of these resources (see this page for what I use in my classroom, and here for even more references), I’m more interested in how smaller problems, things that take students 2-10 minutes to complete, can teach, reinforce, and assess the Practice standards. This was a part of my goal in developing this language for problem design, and it became more explicit as I found more and more ways to think about each of the Practices.
To summarize my thinking, I’m categorizing both my taxonomy of problem design and a number of other resources available around the internet by specific Standard for Mathematical Practice, with the idea that this can serve as a guide to thinking about the different types of understanding students need in order to be proficient mathematicians.
MP.1 – Make sense of problems and persevere in solving them:
Low floor problems, hard problems that are accessible at a range of levels allowing students to fail productively on the way to an answer
One-way problems, problems that are extremely difficult and require thinking beyond what’s expected in a standard, and allow students to synthesize concepts and apply them in a more rigorous way
MP.2 – Reason abstractly and quantitatively:
Reversing, asking students to solve a problem both forwards and backwards to build algebraic thinking
And Fawn’s visual patterns
MP.3 – Construct viable arguments and critique the reasoning of others:
Splitting provoking student discussion by asking a question with several clear possible answers that are in sharp disagreement
MP.4 – Model with mathematics:
Withholding, asking students to identify the information necessary to solve a problem
MP.5 – Use appropriate tools strategically: I have no answers for this one.
MP.6 – Attend to precision:
Thwarting, asking students a question where application of the standard procedure is likely to lead to a wrong answer
Disguised problems, problems that ask students to apply a concept in a format they do not usually see that concept
MP.7 – Look for and make use of structure: I have no answers of my one here, but number talks and counting circles are the two best ideas around
MP.8 – Look for and express regularity in repeated reasoning:
Baiting, providing students an opportunity to see a pattern and use it to solve a problem strategically
Finally, the last two problem types I’ve explored don’t fit neatly into a Practice standard, but are incredibly valuable tools in formative assessment and building understanding of specific concepts –
Jamming, asking a student a question on a concept where the usual procedure cannot be applied
Non-examples, juxtaposing questions that are similar to the standard being taught or assessed but require a different concept to see if students can differentiate between them
That’s what I got.